alnuth-4.0.0/0000755000000000000000000000000015201210200007713 5ustar00alnuth-4.0.0/CHANGES.md0000644000000000000000000000473415201210200011315 0ustar00This file describes changes between different versions of Alnuth ================================================================ ## 4.0.0 (2026-05-14) - added support for using OSCAR as external CAS (in addition to PARI/GP); by default OSCAR support is automatically activated when Alnuth is loaded in GAP running inside OSCAR, in all other situations PARI/GP remains the default. - converted the manual to GAPDoc - renamed `FactorsPolynomialPari` to `FactorsPolynomialAlnuth` to better reflect that it uses the active external CAS, not just PARI/GP (the old name is still supported as a synonym for now) - replaced the old `AL_*` configuration globals by GAP's user preference system; the old variables are still used a fallback, but user preferences are preferred if set. For details, please consult the README and the package manual. ## 3.2.1 (2022-04-05) - set the GAP Team as maintainer of this package - various janitorial changes ## 3.2.0 (2022-03-02) - upgraded to support PARI 2.13 (thanks to Bill Allombert for the patch) ## 3.1.2 (2020-01-28) - updated authors contact data ## 3.1.1 (2019-06-04) - added implications IsUnitGroup => IsGroup and IsNumberFieldByMatrices => IsNumberField for improved compatibility with future GAP releases - fixed a link in the README - removed outdated author contact information ## 3.1.0 (2017-12-01) - upgraded to support PARI 2.9.0 (this required to remove support for PARI < 2.4.3) - migrated the package to GitHub ## 3.0.0 (2011-10-26) - added this file to document the changes between different versions - adjusted documentation to GAP 4.5 - switched interface from KANT/KASH to PARI/GP - replaced all Unix specific code to allow running Alnuth under Windows - removed some errorneous `Set` calls - replaced some calls to `PrimeField` by `Rationals` - removed functions in `unithom.gi` covered by Polycyclic - replaced (possibly expensive) call to `FieldOfPolynomial` - added `SubsetMaintencance` for `IsNumberFieldByMatrices` - renamed `IsPrimitiveElement` to `IsPrimitiveElementOfNumberField` - renamed `IsFieldByMatrices` to `AL_MatricesGeneratingNumberField` ## 2.3.1 (2011-05-31) ## 2.2.5 (2007-07-10) ## 2.2.4 (2007-06-17) ## 2.2.3 (2007-06-11) ## 2.2.2 (2006-11-24) ## 2.2.1 (2006-11-21) ## 2.2.0 (2006-09-02) ## 2.1.3 (2005-08-09) ## 2.1.2 (2005-08-04) ## 2.1.1 (2005-03-10) ## 2.1 (2004-12-20) ## 1.0 (2003-10-09) - initial release alnuth-4.0.0/GPL0000644000000000000000000004311015201210200010257 0ustar00 GNU GENERAL PUBLIC LICENSE Version 2, June 1991 Copyright (C) 1989, 1991 Free Software Foundation, Inc. 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed. 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If this is what you want to do, use the GNU Library General Public License instead of this License. alnuth-4.0.0/Makefile0000644000000000000000000000073215201210200011355 0ustar00.PHONY: run doc html clean check GAP ?= gap GAP_ARGS = -q --quitonbreak --packagedirs $(abspath .) # run GAP and load the package run: $(GAP) --packagedirs $(abspath .) -c 'LoadPackage("alnuth");' doc: $(GAP) $(GAP_ARGS) makedoc.g -c 'QUIT;' html: NOPDF=1 $(GAP) $(GAP_ARGS) makedoc.g -c 'QUIT;' clean: cd doc && rm -f *.{aux,lab,log,dvi,ps,pdf,bbl,ilg,ind,idx,out,html,tex,pnr,txt,blg,toc,six,brf,css,js} _*.xml title.xml check: $(GAP) $(GAP_ARGS) tst/testall.g alnuth-4.0.0/PackageInfo.g0000644000000000000000000001310715201210200012234 0ustar00############################################################################# ## ## PackageInfo.g GAP4 Package "Alnuth" Bjoern Assmann ## Andreas Distler ## SetPackageInfo( rec( PackageName := "Alnuth", Subtitle := "ALgebraic NUmber THeory and an interface to PARI/GP and OSCAR", Version := "4.0.0", Date := "14/05/2026", # dd/mm/yyyy format License := "GPL-2.0-or-later", SourceRepository := rec( Type := "git", URL := Concatenation( "https://github.com/gap-packages/", LowercaseString(~.PackageName) ), ), IssueTrackerURL := Concatenation( ~.SourceRepository.URL, "/issues" ), PackageWWWHome := Concatenation( "https://gap-packages.github.io/", LowercaseString(~.PackageName) ), README_URL := Concatenation( ~.PackageWWWHome, "/README.md" ), PackageInfoURL := Concatenation( ~.PackageWWWHome, "/PackageInfo.g" ), ArchiveURL := Concatenation( ~.SourceRepository.URL, "/releases/download/v", ~.Version, "/", LowercaseString(~.PackageName), "-", ~.Version ), ArchiveFormats := ".tar.gz", Persons := [ rec( LastName := "Assmann", FirstNames := "Björn", IsAuthor := true, IsMaintainer := false, ), rec( LastName := "Distler", FirstNames := "Andreas", IsAuthor := true, IsMaintainer := false, Email := "a.distler@tu-bs.de", ), rec( LastName := "Eick", FirstNames := "Bettina", IsAuthor := true, IsMaintainer := false, Email := "beick@tu-bs.de", WWWHome := "http://www.iaa.tu-bs.de/beick", GitHubUsername:= "beick", PostalAddress := Concatenation( "Institut Analysis und Algebra\n", "TU Braunschweig\n", "Universitätsplatz 2\n", "D-38106 Braunschweig\n", "Germany" ), Place := "Braunschweig", Institution := "TU Braunschweig" ), rec( LastName := "Horn", FirstNames := "Max", IsAuthor := false, IsMaintainer := true, Email := "mhorn@rptu.de", WWWHome := "https://www.quendi.de/math", GitHubUsername:= "fingolfin", PostalAddress := Concatenation( "Fachbereich Mathematik\n", "RPTU Kaiserslautern-Landau\n", "Gottlieb-Daimler-Straße 48\n", "67663 Kaiserslautern\n", "Germany" ), Place := "Kaiserslautern, Germany", Institution := "RPTU Kaiserslautern-Landau" ), rec( LastName := "GAP Team", FirstNames := "The", IsAuthor := false, IsMaintainer := true, Email := "support@gap-system.org", ), ], Status := "accepted", CommunicatedBy := "Charles Wright (Eugene)", AcceptDate := "01/2004", AbstractHTML := "The Alnuth package provides various methods to compute with number fields which are given by a defining polynomial or by generators. Some of the methods provided in this package are written in GAP code. The other part of the methods is imported from an another computer algebra system, currently either PARI/GP or OSCAR. Hence this package contains some GAP functions and an interface to some functions in the computer algebra systems PARI/GP and OSCAR. The main methods included in Alnuth are: creating a number field, computing its maximal order (using the external CAS), computing its unit group (using the external CAS) and a presentation of this unit group, computing the elements of a given norm of the number field (using the external CAS), determining a presentation for a finitely generated multiplicative subgroup (using the external CAS), and factoring polynomials defined over number fields (using the external CAS).", PackageDoc := rec( BookName := "Alnuth", ArchiveURLSubset := ["doc"], HTMLStart := "doc/chap0_mj.html", PDFFile := "doc/manual.pdf", SixFile := "doc/manual.six", LongTitle := "ALgebraic NUmber THeory and an interface to PARI/GP and OSCAR", ), Dependencies := rec( GAP := ">= 4.8", NeededOtherPackages := [[ "polycyclic", ">=1.1" ]], SuggestedOtherPackages := [], NeededSystemPackages := rec( Ubuntu := [["pari-gp"]], Homebrew := [["pari"]] ), ExternalConditions := [["needs the PARI/GP computer algebra system Version 2.5 or higher", "https://pari.math.u-bordeaux.fr/" ] ] ), Extensions := [ rec( needed := [ [ "OscarInterface", ">= 1.0.0" ] ], filename := "gap/oscar.gi" ), ], AvailabilityTest := ReturnTrue, TestFile := "tst/testall.g", Keywords := ["algebraic number theory", "number field" , "maximal order", "interface to PARI/GP", "interface to number theory in OSCAR", "unit group", "elements of given norm" ], AutoDoc := rec( entities := rec( VERSION := ~.Version, RELEASEYEAR := ~.Date{[7..10]}, RELEASEDATE := function(date) local day, month, year, allMonths; day := Int(date{[1,2]}); month := Int(date{[4,5]}); year := Int(date{[7..10]}); allMonths := [ "January", "February", "March", "April", "May", "June", "July", "August", "September", "October", "November", "December"]; return Concatenation(String(day)," ", allMonths[month], " ", String(year)); end(~.Date), Polycyclic := "Polycyclic", ), ), )); alnuth-4.0.0/README.md0000644000000000000000000002132715201210200011177 0ustar00[![CI](https://github.com/gap-packages/alnuth/actions/workflows/CI.yml/badge.svg)](https://github.com/gap-packages/alnuth/actions/workflows/CI.yml) [![Code Coverage](https://codecov.io/github/gap-packages/alnuth/coverage.svg?branch=master&token=)](https://codecov.io/gh/gap-packages/alnuth) # The GAP package Alnuth Alnuth is an extension, a so called package, for the computer algebra system GAP and forms part of a standard installation. For information about GAP see . The functionality of Alnuth lies in ALgebraic NUmber THeory. It provides an interface from GAP to certain number theoretic functions from other computer algebra systems, namely either PARI/GP or OSCAR (in the past also KANT/KASH). Most computations with Alnuth rely on this interface. The interface is an integral part of the package, but the external software (so PARI/GP or OSCAR) has to be obtained independently. ## New in Version 4 Alnuth now has two backends: besides the existing interface to PARI/GP, it now also integrates with the OSCAR computer algebra system when loaded in GAP running inside OSCAR. Moreover many settings in Alnuth that were so far controlled by proprietary controls now use the GAP user preferences system. In particular, the old globals `AL_EXECUTABLE`, `AL_OPTIONS`, `AL_STACKSIZE`, and `PRIM_TEST` are deprecated in favor of `SetUserPreference("alnuth", ...)`, while `AL_PATH` is gone entirely. Deprecated globals still trigger a warning, but except for `AL_PATH` they are used as a fallback if the corresponding new user preference is not set or invalid. ## New in Version 3 Up to and including Version 2.3.1, Alnuth was restricted to operating systems based on Unix. This is no longer the case and Alnuth can now also be used under Windows. Moreover, former versions of Alnuth provided an interface to KANT, respectively its shell KASH, instead of one to PARI/GP. This change did not influence the availability of GAP functions in the package. Note that any further changes and bugfixes will only be made to Version 3 of Alnuth which contains the interface to PARI/GP. ## Installing Alnuth The package Alnuth is part of the standard distribution of GAP so that in most cases there is no need to install it separately. If you are using OSCAR, then Alnuth is already bundled and set up to interface with OSCAR, and you can stop reading this section. Otherwise, if you are using it outside of OSCAR, you need to have PARI/GP installed. See the following section for information on PARI/GP. In case you want to update Alnuth independently of your main GAP installation, you can download it from its homepage . If you are interested in an old version of Alnuth interfacing to KANT/KASH you can find all released versions of Alnuth starting from v1.0 (09/10/2003) at . There are two ways of installing a GAP package. If you have permission to add files to the installation of GAP on your system you may install Alnuth into the `pkg` subdirectory of the GAP installation tree. Otherwise you may install Alnuth in a private `pkg` directory (for details see '76.1 Installing a GAP Package' and '9.2 GAP Root Directories' in the GAP reference manual). To install the latest version of Alnuth download the `alnuth.tar.gz` archive, move it to the directory `pkg` in which you want to install, and unpack the archive. If you are using the command line you can unpack with the command `tar xzf alnuth.tar.gz`. ### Getting PARI/GP Using Alnuth outside of OSCAR requires an installation of PARI/GP Version 2.5 or higher. PARI/GP is freely available at . Note that the place where PARI/GP is located in your system is independent of the place where Alnuth is installed. In many Linux distributions PARI/GP can be installed via the software package manager, but this might sometimes be an older version which cannot be used together with Alnuth. If you install PARI/GP from source make sure you install with GMP support for better performance and complete the installation with `make install` so that you can start GP by just calling `gp` from the command line. For Windows it is sufficient to get the basic GP binary which can be found at . ### Adjust the path of the executable for GP This package needs to know where the executable for GP is. In the default setting Alnuth looks for an executable program named `gp` in the search paths of the system. More precisely, for a file `gp` inside one of the directories in the list returned by `DirectoriesSystemPrograms()`. Under Linux the default setting should work with a standard installation of PARI/GP. For the default setting to work under Windows the downloaded executable file, for example `gp-2-5-0.exe` has to be renamed to `gp.exe` and moved to one of the directories listed by `DirectoriesSystemPrograms()`. If you cannot use the default setting for you purpose, you can find more information in the last chapter of the Alnuth manual. The most important user preferences are: - `SetUserPreference("alnuth", "PariGpPath", "/path/to/gp");` - `SetUserPreference("alnuth", "PariStackSize", 256);` - `SetUserPreference("alnuth", "PrimitiveElementTrials", 40);` Then call `WriteGapIniFile();` to write the current non-default settings to your `gap.ini`. Users upgrading from older Alnuth versions should note these replacements: - `AL_EXECUTABLE` becomes `PariGpPath` - `AL_OPTIONS` becomes `PariGpOptions` - `AL_STACKSIZE` becomes `PariStackSize` - `PRIM_TEST` becomes `PrimitiveElementTrials` - `AL_PATH` no longer exists; Alnuth now locates its bundled GP code itself The deprecated globals still act as a fallback if the corresponding user preference is unusable, but they trigger a warning and should be replaced. The old functions `SetAlnuthExternalExecutable` and `SetAlnuthExternalExecutablePermanently` still exist as deprecated compatibility wrappers around the new preferences. ## Loading the package If Alnuth is not loaded when GAP is started you have to request it explicitly to use it. This is done by calling `LoadPackage("Alnuth");` in a GAP session. If Alnuth had not been loaded already a short banner will be displayed. gap> LoadPackage("Alnuth"); Loading Alnuth 4.0.0 (ALgebraic NUmber THeory and an interface to PARI/GP and OSCAR) by Björn Assmann, Andreas Distler (a.distler@tu-bs.de), and Bettina Eick (http://www.iaa.tu-bs.de/beick). maintained by: Max Horn (https://www.quendi.de/math) and The GAP Team (support@gap-system.org). Homepage: https://gap-packages.github.io/alnuth Report issues at https://github.com/gap-packages/alnuth/issues true gap> To load a certain version of Alnuth you can specify the version number as second argument in the call to `LoadPackage`. (See '76.2 Loading a GAP package' in the reference manual or type `?LoadPackage` within a GAP session). ## Testing the package Once the package is loaded, it is possible to check the correct installation running a short test by calling `ReadPackage("Alnuth", "tst/testinstall.g");`. gap> ReadPackage("Alnuth", "tst/testinstall.g"); Architecture: aarch64-apple-darwin24-default64-kv10 testing: GAPROOT/pkg/alnuth/tst/ALNUTH.tst 72 ms (41 ms GC) and 10.9MB allocated for GAPROOT/pkg/alnuth/tst/ALNUTH.tst ----------------------------------- total 72 ms (41 ms GC) and 10.9MB allocated 0 failures in 1 files #I No errors detected while testing If the test suite runs into an error in the first part, which verifies the availability of PARI/GP, check your installation of PARI/GP and consult the last chapter of the documentation of Alnuth for more information. ## Contact If you find any bugs or have any suggestions or comments, we would very much appreciate it if you would let us know by submitting an issue at the Alnuth issue tracker on GitHub or by writing an mail to . ## License > Copyright (C) 2011 Bjoern Assmann, Andreas Distler, Bettina Eick > > This program is free software: you can redistribute it and/or modify > it under the terms of the GNU General Public License as published by > the Free Software Foundation, either version 2 of the license, or > (at your option) any later version. > > This program is distributed in the hope that it will be useful, > but WITHOUT ANY WARRANTY; without even the implied warranty of > MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the > GNU General Public License for more details. > > You should have received a copy of the GNU General Public License > along with this program. If not, see . alnuth-4.0.0/doc/0000755000000000000000000000000015201210200010460 5ustar00alnuth-4.0.0/doc/_AutoDocMainFile.xml0000644000000000000000000000014115201210200014300 0ustar00   alnuth-4.0.0/doc/_Chunks.xml0000644000000000000000000000000015201210200012562 0ustar00alnuth-4.0.0/doc/_entities.xml0000644000000000000000000000027515201210200013171 0ustar00Alnuth'> Polycyclic'> alnuth-4.0.0/doc/_main.tex0000644000000000000000000012444615201210200012300 0ustar00% generated by GAPDoc2LaTeX from XML source (Frank Luebeck) \documentclass[a4paper,11pt]{report} \usepackage[top=37mm,bottom=37mm,left=27mm,right=27mm]{geometry} \sloppy \pagestyle{myheadings} \usepackage{amssymb} \usepackage[utf8]{inputenc} \usepackage{makeidx} \makeindex \usepackage{color} \definecolor{FireBrick}{rgb}{0.5812,0.0074,0.0083} \definecolor{RoyalBlue}{rgb}{0.0236,0.0894,0.6179} \definecolor{RoyalGreen}{rgb}{0.0236,0.6179,0.0894} \definecolor{RoyalRed}{rgb}{0.6179,0.0236,0.0894} \definecolor{LightBlue}{rgb}{0.8544,0.9511,1.0000} \definecolor{Black}{rgb}{0.0,0.0,0.0} \definecolor{linkColor}{rgb}{0.0,0.0,0.554} \definecolor{citeColor}{rgb}{0.0,0.0,0.554} \definecolor{fileColor}{rgb}{0.0,0.0,0.554} \definecolor{urlColor}{rgb}{0.0,0.0,0.554} \definecolor{promptColor}{rgb}{0.0,0.0,0.589} \definecolor{brkpromptColor}{rgb}{0.589,0.0,0.0} \definecolor{gapinputColor}{rgb}{0.589,0.0,0.0} \definecolor{gapoutputColor}{rgb}{0.0,0.0,0.0} %% for a long time these were red and blue by default, %% now black, but keep variables to overwrite \definecolor{FuncColor}{rgb}{0.0,0.0,0.0} %% strange name because of pdflatex bug: \definecolor{Chapter }{rgb}{0.0,0.0,0.0} \definecolor{DarkOlive}{rgb}{0.1047,0.2412,0.0064} \usepackage{fancyvrb} \usepackage{mathptmx,helvet} \usepackage[T1]{fontenc} \usepackage{textcomp} \usepackage[ pdftex=true, bookmarks=true, a4paper=true, pdftitle={Written with GAPDoc}, pdfcreator={LaTeX with hyperref package / GAPDoc}, colorlinks=true, backref=page, breaklinks=true, linkcolor=linkColor, citecolor=citeColor, filecolor=fileColor, urlcolor=urlColor, pdfpagemode={UseNone}, ]{hyperref} \newcommand{\maintitlesize}{\fontsize{50}{55}\selectfont} % write page numbers to a .pnr log file for online help \newwrite\pagenrlog \immediate\openout\pagenrlog =\jobname.pnr \immediate\write\pagenrlog{PAGENRS := [} \newcommand{\logpage}[1]{\protect\write\pagenrlog{#1, \thepage,}} %% were never documented, give conflicts with some additional packages \newcommand{\GAP}{\textsf{GAP}} %% nicer description environments, allows long labels \usepackage{enumitem} \setdescription{style=nextline} %% depth of toc \setcounter{tocdepth}{1} %% command for ColorPrompt style examples \newcommand{\gapprompt}[1]{\color{promptColor}{\bfseries #1}} \newcommand{\gapbrkprompt}[1]{\color{brkpromptColor}{\bfseries #1}} \newcommand{\gapinput}[1]{\color{gapinputColor}{#1}} \begin{document} \logpage{[ 0, 0, 0 ]} \begin{titlepage} \mbox{}\vfill \begin{center}{\maintitlesize \textbf{ Alnuth \\ \mbox{}}}\\ \vfill \hypersetup{pdftitle={ Alnuth }} \markright{\scriptsize \mbox{}\hfill Alnuth \hfill\mbox{}} {\Huge \textbf{ ALgebraic NUmber THeory\\ and an interface to\\ PARI/GP and OSCAR \\ \mbox{}}}\\ \vfill {\Huge 4.0.0 \mbox{}}\\[1cm] { 14 May 2026 \mbox{}}\\[1cm] \mbox{}\\[2cm] {\Large \textbf{\strut GAP code by Bj{\"o}rn Assmann, Andreas Distler and Bettina Eick \strut\mbox{}}}\\ {\Large \textbf{\strut PARI/GP code by Bill Allombert \strut\mbox{}}}\\ {\Large \textbf{\strut OSCAR code by Claus Fieker and Max Horn \strut\mbox{}}}\\ \hypersetup{pdfauthor={ GAP code by Bj{\"o}rn Assmann, Andreas Distler and Bettina Eick ; PARI/GP code by Bill Allombert ; OSCAR code by Claus Fieker and Max Horn }} \mbox{}\\[2cm] \begin{minipage}{12cm}\noindent \emph{Note:} PARI/GP is \emph{not} part of this package. It can be obtained from \href{https://pari.math.u-bordeaux.fr/} {\texttt{https://pari.math.u\texttt{\symbol{45}}bordeaux.fr/}}. If you use GAP via OSCAR, then OSCAR will automatically be used instead of PARI/GP. \end{minipage} \end{center}\vfill \mbox{}\\ \end{titlepage} \newpage\setcounter{page}{2} {\small \section*{Copyright} \logpage{[ 0, 0, 1 ]} This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 2 of the license, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program. If not, see \href{https://www.gnu.org/licenses/} {\texttt{https://www.gnu.org/licenses/}} \mbox{}}\\[1cm] {\small \section*{Acknowledgements} \logpage{[ 0, 0, 2 ]} To begin with we are very grateful for all the feedback by users of former versions of \textsf{Alnuth}. We thank Bill Allombert who wrote the GP code for the interface to PARI/GP and who was extremely helpful in the transition from KANT to PARI/GP. For feedback on the development version, including a code patch, we are much obliged to Max Horn. The second author acknowledges the financial support at CAUL within the projects PTDC/MAT/101993/2008 and ISFL\texttt{\symbol{45}}1\texttt{\symbol{45}}143, financed by FEDER and FCT, in the development of Version 3 of \textsf{Alnuth}. Support for using OSCAR instead of PARI/GP was added by Claus Fieker and Max Horn. \mbox{}}\\[1cm] \newpage \def\contentsname{Contents\logpage{[ 0, 0, 3 ]}} \tableofcontents \newpage \chapter{\textcolor{Chapter }{Introduction}}\label{Introduction} \logpage{[ 1, 0, 0 ]} \hyperdef{L}{X7DFB63A97E67C0A1}{} { A number field is a finite extension of the field of rational numbers. \textsf{Alnuth} provides various methods to compute with number fields which are given by a defining polynomial or by generators. For background on number fields we refer to \cite{Sta79}. Some of the methods provided in this package are written in \textsf{GAP} code. The other part of the methods is imported from the Computer Algebra Systems PARI/GP \cite{PARI2} respectively OSCAR \cite{Oscar25}, \cite{OSCAR}. Hence this package contains some \textsf{GAP} functions and an interface to some functions to these computer algebra systems. Therefore one has to have PARI/GP or OSCAR installed to use the full functionality of \textsf{Alnuth}. We note that only a very small part of the functions available in PARI/GP respectively OSCAR are linked to \textsf{GAP} and they provides many more methods for computations in number fields. The main methods included in \textsf{Alnuth} are: creating a number field, computing its maximal order, computing its unit group and a presentation of this unit group, computing the elements of a given norm of the number field, determining a presentation for a finitely generated multiplicative subgroup, and factoring polynomials defined over number fields. For background on algorithms for number fields we refer to \cite{Poh93}, \cite{PZa89} and \cite{Coh93}. The functions provided by \textsf{Alnuth} are introduced in the following chapter. Then an example application is outlined. In the final chapter of this manual the installation of the package and configuration of the interface, including hints on the installation of PARI/GP respectively OSCAR, are described. } \chapter{\textcolor{Chapter }{Working with number fields}}\label{Working with number fields} \logpage{[ 2, 0, 0 ]} \hyperdef{L}{X7DC995CB8412F8E2}{} { An algebraic number field is a finite\texttt{\symbol{45}}dimensional extension of the rational numbers ${\ensuremath{\mathbb Q}}$. Such a number field has a primitive element and it can be defined by the minimal polynomial of this primitive element. Another important way to define an algebraic number field is by a set of rational matrices which generate a number field. \section{\textcolor{Chapter }{Creation of number fields}}\label{Creation of number fields} \logpage{[ 2, 1, 0 ]} \hyperdef{L}{X86AD0C3F7D38C269}{} { We provide functions to create number fields defined by rational matrices or by rational polynomials. \subsection{\textcolor{Chapter }{FieldByMatricesNC}} \logpage{[ 2, 1, 1 ]}\nobreak \hyperdef{L}{X8085204B7DE2C8BA}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{FieldByMatricesNC({\mdseries\slshape matrices})\index{FieldByMatricesNC@\texttt{FieldByMatricesNC}} \label{FieldByMatricesNC} }\hfill{\scriptsize (function)}}\\ \noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{FieldByMatrices({\mdseries\slshape matrices})\index{FieldByMatrices@\texttt{FieldByMatrices}} \label{FieldByMatrices} }\hfill{\scriptsize (function)}}\\ Creates a field generated by the rational matrices \mbox{\texttt{\mdseries\slshape matrices}}. In the faster NC version, the function assumes that the input generates a field and there are no checks on this performed. } \subsection{\textcolor{Chapter }{FieldByMatrixBasisNC}} \logpage{[ 2, 1, 2 ]}\nobreak \hyperdef{L}{X805CFBEB877FCBA3}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{FieldByMatrixBasisNC({\mdseries\slshape matrices})\index{FieldByMatrixBasisNC@\texttt{FieldByMatrixBasisNC}} \label{FieldByMatrixBasisNC} }\hfill{\scriptsize (function)}}\\ \noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{FieldByMatrixBasis({\mdseries\slshape matrices})\index{FieldByMatrixBasis@\texttt{FieldByMatrixBasis}} \label{FieldByMatrixBasis} }\hfill{\scriptsize (function)}}\\ Creates a field with basis \mbox{\texttt{\mdseries\slshape matrices}}. The list \mbox{\texttt{\mdseries\slshape matrices}} must consist of rational matrices which form a basis for a number field. In the faster NC version, the function assumes that the input is a matrix basis for a field and no checks are performed. } \subsection{\textcolor{Chapter }{FieldByPolynomialNC}} \logpage{[ 2, 1, 3 ]}\nobreak \hyperdef{L}{X7AD9823687E61ABC}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{FieldByPolynomialNC({\mdseries\slshape polynomial})\index{FieldByPolynomialNC@\texttt{FieldByPolynomialNC}} \label{FieldByPolynomialNC} }\hfill{\scriptsize (function)}}\\ \noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{FieldByPolynomial({\mdseries\slshape polynomial})\index{FieldByPolynomial@\texttt{FieldByPolynomial}} \label{FieldByPolynomial} }\hfill{\scriptsize (function)}}\\ Creates a field defined by \mbox{\texttt{\mdseries\slshape polynomial}}. The polynomial \mbox{\texttt{\mdseries\slshape polynomial}} must be an irreducible rational polynomial. In the faster NC version, no checks on the input are performed. } } \section{\textcolor{Chapter }{Methods for number fields}}\label{Methods for number fields} \logpage{[ 2, 2, 0 ]} \hyperdef{L}{X79007A477D53B572}{} { We outline a number of functions for number fields. \subsection{\textcolor{Chapter }{PrimitiveElement}} \logpage{[ 2, 2, 1 ]}\nobreak \hyperdef{L}{X86DB31B57FB4F570}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{PrimitiveElement({\mdseries\slshape F})\index{PrimitiveElement@\texttt{PrimitiveElement}} \label{PrimitiveElement} }\hfill{\scriptsize (function)}}\\ \noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{DefiningPolynomial({\mdseries\slshape F})\index{DefiningPolynomial@\texttt{DefiningPolynomial}} \label{DefiningPolynomial} }\hfill{\scriptsize (function)}}\\ Computes a primitive element and a defining polynomial for the given number field. The defining polynomial is the minimal polynomial of the primitive element. Since \mbox{\texttt{\mdseries\slshape F}} contains various primitive elements, \texttt{PrimitiveElement} tries to find a primitive element which has a minimal polynomial with small coefficients. Via the user preference \texttt{PrimitiveElementTrials} the user can decide how many primitive elements will be compared. The default value is 20. (See \texttt{SetUserPreference} (\textbf{Reference: SetUserPreference}) for more information about user preferences.) } \subsection{\textcolor{Chapter }{IsPrimitiveElementOfNumberField}} \logpage{[ 2, 2, 2 ]}\nobreak \hyperdef{L}{X843C2BE881A3576A}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{IsPrimitiveElementOfNumberField({\mdseries\slshape F, a})\index{IsPrimitiveElementOfNumberField@\texttt{IsPrimitiveElementOfNumberField}} \label{IsPrimitiveElementOfNumberField} }\hfill{\scriptsize (function)}}\\ Checks if the given element generates the field. } \subsection{\textcolor{Chapter }{DegreeOverPrimeField}} \logpage{[ 2, 2, 3 ]}\nobreak \hyperdef{L}{X7845CECE86A83219}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{DegreeOverPrimeField({\mdseries\slshape F})\index{DegreeOverPrimeField@\texttt{DegreeOverPrimeField}} \label{DegreeOverPrimeField} }\hfill{\scriptsize (function)}}\\ Returns the degree of \mbox{\texttt{\mdseries\slshape F}} over the rationals. } \subsection{\textcolor{Chapter }{EquationOrderBasis}} \logpage{[ 2, 2, 4 ]}\nobreak \hyperdef{L}{X8185FEF1811EA43F}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{EquationOrderBasis({\mdseries\slshape F})\index{EquationOrderBasis@\texttt{EquationOrderBasis}} \label{EquationOrderBasis} }\hfill{\scriptsize (function)}}\\ \noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{MaximalOrderBasis({\mdseries\slshape F})\index{MaximalOrderBasis@\texttt{MaximalOrderBasis}} \label{MaximalOrderBasis} }\hfill{\scriptsize (function)}}\\ \noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{IsIntegerOfNumberField({\mdseries\slshape F, k})\index{IsIntegerOfNumberField@\texttt{IsIntegerOfNumberField}} \label{IsIntegerOfNumberField} }\hfill{\scriptsize (function)}}\\ These functions return bases for the equation order or the maximal order of the number field \mbox{\texttt{\mdseries\slshape F}}. Also, they allow to check if a given element is an integer in the given number field. } \subsection{\textcolor{Chapter }{UnitGroup}} \logpage{[ 2, 2, 5 ]}\nobreak \hyperdef{L}{X7902A7877B65FDA1}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{UnitGroup({\mdseries\slshape F})\index{UnitGroup@\texttt{UnitGroup}} \label{UnitGroup} }\hfill{\scriptsize (function)}}\\ determines the unit group of \mbox{\texttt{\mdseries\slshape F}}. Recall that the unit group of \mbox{\texttt{\mdseries\slshape F}} is a finitely generated abelian group. The function \texttt{IsomorphismPcpGroup} from the \textsf{Polycyclic} \cite{Polycyclic} package gives an isomorphism to a pcp group which can be used for various computations with the unit group. } \subsection{\textcolor{Chapter }{IsUnitOfNumberField}} \logpage{[ 2, 2, 6 ]}\nobreak \hyperdef{L}{X7D9484BA7AF4201C}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{IsUnitOfNumberField({\mdseries\slshape F, k})\index{IsUnitOfNumberField@\texttt{IsUnitOfNumberField}} \label{IsUnitOfNumberField} }\hfill{\scriptsize (function)}}\\ checks whether the element \mbox{\texttt{\mdseries\slshape k}} is a unit in \mbox{\texttt{\mdseries\slshape F}}. } \subsection{\textcolor{Chapter }{ExponentsOfUnits}} \logpage{[ 2, 2, 7 ]}\nobreak \hyperdef{L}{X80389D828474FC64}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{ExponentsOfUnits({\mdseries\slshape F, elms})\index{ExponentsOfUnits@\texttt{ExponentsOfUnits}} \label{ExponentsOfUnits} }\hfill{\scriptsize (function)}}\\ This function determines the exponent vectors of the elements in \mbox{\texttt{\mdseries\slshape elms}} with respect to the generators of the unit group of \mbox{\texttt{\mdseries\slshape F}}. If the unit group of \mbox{\texttt{\mdseries\slshape F}} is not known, then the function computes this unit group also. } \subsection{\textcolor{Chapter }{IsCyclotomicField}} \logpage{[ 2, 2, 8 ]}\nobreak \hyperdef{L}{X84CAE4627F0CD639}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{IsCyclotomicField({\mdseries\slshape F})\index{IsCyclotomicField@\texttt{IsCyclotomicField}} \label{IsCyclotomicField} }\hfill{\scriptsize (function)}}\\ Check whether \mbox{\texttt{\mdseries\slshape F}} is cyclotomic. } \subsection{\textcolor{Chapter }{NormCosetsOfNumberField}} \logpage{[ 2, 2, 9 ]}\nobreak \hyperdef{L}{X7A43318584DB3756}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{NormCosetsOfNumberField({\mdseries\slshape F, norm})\index{NormCosetsOfNumberField@\texttt{NormCosetsOfNumberField}} \label{NormCosetsOfNumberField} }\hfill{\scriptsize (function)}}\\ Returns a description for the set of all elements of norm \mbox{\texttt{\mdseries\slshape norm}} in \mbox{\texttt{\mdseries\slshape F}}. These elements can be written as a finite union of cosets of the unit group of \mbox{\texttt{\mdseries\slshape F}}. The function returns coset representatives for these cosets. } } \section{\textcolor{Chapter }{Presentations of multiplicative subgroups}}\label{Presentations of multiplicative subgroups} \logpage{[ 2, 3, 0 ]} \hyperdef{L}{X833A0296798E0463}{} { Suppose that a finite number of invertible elements of a number field are given. Then these elements generate a finitely generated abelian group. However, it is a non\texttt{\symbol{45}}trivial task to provide a presentation for this abelian group. The most useful representation for such groups is as pcp group. \subsection{\textcolor{Chapter }{PcpPresentationOfMultiplicativeSubgroup}} \logpage{[ 2, 3, 1 ]}\nobreak \hyperdef{L}{X7975303583E6058B}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{PcpPresentationOfMultiplicativeSubgroup({\mdseries\slshape F, elms})\index{PcpPresentationOfMultiplicativeSubgroup@\texttt{Pcp}\-\texttt{Presentation}\-\texttt{Of}\-\texttt{Multiplicative}\-\texttt{Subgroup}} \label{PcpPresentationOfMultiplicativeSubgroup} }\hfill{\scriptsize (function)}}\\ \noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{IsomorphismPcpGroup({\mdseries\slshape F, elms})\index{IsomorphismPcpGroup@\texttt{IsomorphismPcpGroup}} \label{IsomorphismPcpGroup} }\hfill{\scriptsize (function)}}\\ Determine a pcp presentation for the multiplicative group of $\mbox{\texttt{\mdseries\slshape F}}\backslash\{0\}$ generated by \mbox{\texttt{\mdseries\slshape elms}} and an isomorphism on this presentation. Note, that the method \texttt{IsomorphismPcpGroup} is defined in the \textsf{Polycyclic} package \cite{Polycyclic}. We refer to the manual of this package for further background. In the determination of the Pcp\texttt{\symbol{45}}presentation of a multiplicative subgroup generated by \mbox{\texttt{\mdseries\slshape elms}} the relations between the elements in \mbox{\texttt{\mdseries\slshape elms}} play an important role. Let $elms=\{e_1,\dots,e_l\}$ be a finite subset of a field \mbox{\texttt{\mdseries\slshape F}}. The relation lattice for \mbox{\texttt{\mdseries\slshape elms}} is \[ rl(elms):=\left\{(h_1,\dots,h_l) \in {\ensuremath{\mathbb Z}}^l | e_1^{h_1} \cdots e_l^{h_l} = 1\right\} . \] } \subsection{\textcolor{Chapter }{RelationLattice}} \logpage{[ 2, 3, 2 ]}\nobreak \hyperdef{L}{X7C6E88FE82BB9DE3}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{RelationLattice({\mdseries\slshape F, elms})\index{RelationLattice@\texttt{RelationLattice}} \label{RelationLattice} }\hfill{\scriptsize (function)}}\\ Determines a generating set for the relation lattice of the field elements \mbox{\texttt{\mdseries\slshape elms}}. } } \section{\textcolor{Chapter }{Methods to compute with subgroups of the unit group}}\label{Methods to compute with subgroups of the unit group} \logpage{[ 2, 4, 0 ]} \hyperdef{L}{X86F68B4883142B5A}{} { \subsection{\textcolor{Chapter }{RelationLatticeOfUnits}} \logpage{[ 2, 4, 1 ]}\nobreak \hyperdef{L}{X82B056EC85CF150D}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{RelationLatticeOfUnits({\mdseries\slshape F, elms})\index{RelationLatticeOfUnits@\texttt{RelationLatticeOfUnits}} \label{RelationLatticeOfUnits} }\hfill{\scriptsize (function)}}\\ Determines a basis for the relation lattice of the units \mbox{\texttt{\mdseries\slshape elms}} in triangularized form. Note that this method is more efficient than the method \texttt{RelationLattice}. } \subsection{\textcolor{Chapter }{IntersectionOfUnitSubgroups}} \logpage{[ 2, 4, 2 ]}\nobreak \hyperdef{L}{X7B8C89C980CB15F7}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{IntersectionOfUnitSubgroups({\mdseries\slshape F, gen1, gen2})\index{IntersectionOfUnitSubgroups@\texttt{IntersectionOfUnitSubgroups}} \label{IntersectionOfUnitSubgroups} }\hfill{\scriptsize (function)}}\\ The lists \mbox{\texttt{\mdseries\slshape gen1}} and \mbox{\texttt{\mdseries\slshape gen2}} are supposed to generate two subgroups $U_1$ and $U_2$ of the unit group of \mbox{\texttt{\mdseries\slshape F}}. This function determines the intersection of $U_1$ with $U_2$. The result is returned as a list of vectors generating the lattice $\{ e \in {\ensuremath{\mathbb Z}}^n \mid g_1^{e_1} \cdots g_n^{e_n} \in U_2 \}$ for \mbox{\texttt{\mdseries\slshape gen1}} = $[g_1, \ldots, g_n]$. For efficiency reasons this function does not check the input and it may return wrong results if the input generators do not fulfil the requirements. } } \section{\textcolor{Chapter }{Factorisation of polynomials over a number field}}\label{Factorisation of polynomials over a number field} \logpage{[ 2, 5, 0 ]} \hyperdef{L}{X7F656B217EF114DA}{} { \subsection{\textcolor{Chapter }{FactorsPolynomialAlgExt}} \logpage{[ 2, 5, 1 ]}\nobreak \hyperdef{L}{X815A5FBE805119CC}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{FactorsPolynomialAlgExt({\mdseries\slshape F, pol})\index{FactorsPolynomialAlgExt@\texttt{FactorsPolynomialAlgExt}} \label{FactorsPolynomialAlgExt} }\hfill{\scriptsize (function)}}\\ embeds the rational polynomial \mbox{\texttt{\mdseries\slshape pol}} into the polynomial ring over the number field \mbox{\texttt{\mdseries\slshape F}}, which has to be constructed by \texttt{FieldByPolynomial} or \texttt{AlgebraicExtension}, and returns the factorization of the embedded polynomial. By default \mbox{\texttt{\mdseries\slshape a}} denotes the primitive element of the field one can obtain from \texttt{PrimitiveElement(\mbox{\texttt{\mdseries\slshape F}})}, that is, a root of the defining polynomial of \mbox{\texttt{\mdseries\slshape F}}. } \subsection{\textcolor{Chapter }{FactorsPolynomialAlnuth}} \logpage{[ 2, 5, 2 ]}\nobreak \hyperdef{L}{X7CC3F0458523CB22}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{FactorsPolynomialAlnuth({\mdseries\slshape pol})\index{FactorsPolynomialAlnuth@\texttt{FactorsPolynomialAlnuth}} \label{FactorsPolynomialAlnuth} }\hfill{\scriptsize (function)}}\\ takes a polynomial \mbox{\texttt{\mdseries\slshape pol}} defined over an algebraic extension of the Rationals and factors it using the external CAS (PARI/GP or OSCAR). The alias \texttt{FactorsPolynomialPari} is provided for backwards compatibility. \begin{Verbatim}[commandchars=@|C,fontsize=\small,frame=single,label=Example] @gapprompt|gap>C @gapinput|x := Indeterminate( Rationals, "x" );;C @gapprompt|gap>C @gapinput|pol := 2*x^7+2*x^5+8*x^4+8*x^2;C 2*x^7+2*x^5+8*x^4+8*x^2 @gapprompt|gap>C @gapinput|L := FieldByPolynomial( x^3-4 );C @gapprompt|gap>C @gapinput|y := Indeterminate( L, "y" );;C @gapprompt|gap>C @gapinput|FactorsPolynomialAlgExt( L, pol );C [ !2*y, y, y+a, y^2+!1, y^2+(-a)*y+a^2 ] @gapprompt|gap>C @gapinput|FactorsPolynomialAlnuth( last[5] );C [ y^2+(-a)*y+a^2 ] \end{Verbatim} } } \section{\textcolor{Chapter }{Examples}}\label{Examples} \logpage{[ 2, 6, 0 ]} \hyperdef{L}{X7A489A5D79DA9E5C}{} { \subsection{\textcolor{Chapter }{ExampleMatField}} \logpage{[ 2, 6, 1 ]}\nobreak \hyperdef{L}{X7B4AE27B7BB8A9ED}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{ExampleMatField({\mdseries\slshape l})\index{ExampleMatField@\texttt{ExampleMatField}} \label{ExampleMatField} }\hfill{\scriptsize (function)}}\\ This function returns some examples of fields generated by matrices. There are 9 such example fields provided and they can be obtained by assigning the input \mbox{\texttt{\mdseries\slshape l}} to an integer between 1 and 9. Some of the properties of the examples are summarized in the following table. \begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] degree over Q number of generators dim. of generators ExampleMatField(1) 4 4 4 ExampleMatField(2) 4 4 4 ExampleMatField(3) 4 4 4 ExampleMatField(4) 4 13 4 ExampleMatField(5) 4 13 4 ExampleMatField(6) 4 7 4 ExampleMatField(7) 4 18 4 ExampleMatField(8) 4 13 4 ExampleMatField(9) 4 7 4 \end{Verbatim} } } } \chapter{\textcolor{Chapter }{An example application}}\label{An example application} \logpage{[ 3, 0, 0 ]} \hyperdef{L}{X81CAD2F27B2066C4}{} { In this section we outline two example computations with the functions of the previous chapter. The first example uses number fields defined by matrices and the second example considers number fields defined by a polynomial. \section{\textcolor{Chapter }{Number fields defined by matrices}}\label{Number fields defined by matrices} \logpage{[ 3, 1, 0 ]} \hyperdef{L}{X7981DB6C79742C4C}{} { \begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] !gapprompt@gap>| !gapinput@m1 := [ [ 1, 0, 0, -7 ], | !gapprompt@>| !gapinput@ [ 7, 1, 0, -7 ], | !gapprompt@>| !gapinput@ [ 0, 7, 1, -7 ], | !gapprompt@>| !gapinput@ [ 0, 0, 7, -6 ] ];;| # !gapprompt@gap>| !gapinput@m2 := [ [ 0, 0, -13, 14 ], | !gapprompt@>| !gapinput@ [ -1, 0, -13, 1 ], | !gapprompt@>| !gapinput@ [ 13, -1, -13, 1 ], | !gapprompt@>| !gapinput@ [ 0, 13, -14, 1 ] ];;| # !gapprompt@gap>| !gapinput@F := FieldByMatricesNC( [m1, m2] );| # !gapprompt@gap>| !gapinput@DegreeOverPrimeField(F);| 4 !gapprompt@gap>| !gapinput@PrimitiveElement(F);| [ [ 0, -1, 1, 0 ], [ 0, -1, 0, 1 ], [ 0, -1, 0, 0 ], [ 1, -1, 0, 0 ] ] # !gapprompt@gap>| !gapinput@Basis(F);| Basis( , [ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 0, 1, 0, 0 ], [ -1, 1, 1, 0 ], [ -1, 0, 1, 1 ], [ -1, 0, 0, 1 ] ], [ [ 0, 0, 1, 0 ], [ -1, 0, 1, 1 ], [ -1, -1, 1, 1 ], [ 0, -1, 0, 1 ] ], [ [ 0, 0, 0, 1 ], [ -1, 0, 0, 1 ], [ 0, -1, 0, 1 ], [ 0, 0, -1, 1 ] ] ] ) # !gapprompt@gap>| !gapinput@MaximalOrderBasis(F);| Basis( , [ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 0, -1, 1, 0 ], [ 0, -1, 0, 1 ], [ 0, -1, 0, 0 ], [ 1, -1, 0, 0 ] ], [ [ 0, 0, 0, -1 ], [ 1, 0, 0, -1 ], [ 0, 1, 0, -1 ], [ 0, 0, 1, -1 ] ], [ [ -1, 1, 0, 0 ], [ -1, 0, 1, 0 ], [ -1, 0, 0, 1 ], [ -1, 0, 0, 0 ] ] ] ) # !gapprompt@gap>| !gapinput@U := UnitGroup(F);| # !gapprompt@gap>| !gapinput@u := GeneratorsOfGroup( U );;| # !gapprompt@gap>| !gapinput@nat := IsomorphismPcpGroup(U);;| !gapprompt@gap>| !gapinput@H := Image(nat);| Pcp-group with orders [ 10, 0 ] !gapprompt@gap>| !gapinput@ImageElm( nat, u[1] );| g1 !gapprompt@gap>| !gapinput@ImageElm( nat, u[2] );| g2 !gapprompt@gap>| !gapinput@ImageElm( nat, u[1]*u[2] );| g1*g2 !gapprompt@gap>| !gapinput@u[1] = PreImagesRepresentative(nat, GeneratorsOfGroup(H)[1] );| true \end{Verbatim} } \section{\textcolor{Chapter }{Number fields defined by a polynomial}}\label{Number fields defined by a polynomial} \logpage{[ 3, 2, 0 ]} \hyperdef{L}{X85ADC6A27B26C804}{} { \begin{Verbatim}[commandchars=@|D,fontsize=\small,frame=single,label=Example] @gapprompt|gap>D @gapinput|g := UnivariatePolynomial( Rationals, [ 16, 64, -28, -4, 1 ] );D x^4-4*x^3-28*x^2+64*x+16 # @gapprompt|gap>D @gapinput|F := FieldByPolynomialNC(g);D @gapprompt|gap>D @gapinput|PrimitiveElement(F);D a @gapprompt|gap>D @gapinput|MaximalOrderBasis(F);D Basis( , [ !1, 1/2*a, 1/4*a^2, 1/56*a^3+1/14*a^2+1/14*a-2/7 ] ) # @gapprompt|gap>D @gapinput|U := UnitGroup(F);D # @gapprompt|gap>D @gapinput|natU := IsomorphismPcpGroup(U);;D @gapprompt|gap>D @gapinput|elms := List( [1..10], x-> Random(F) );;D # @gapprompt|gap>D @gapinput| PcpPresentationOfMultiplicativeSubgroup( F, elms );D Pcp-group with orders [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] # @gapprompt|gap>D @gapinput|isom := IsomorphismPcpGroup( F, elms );;D @gapprompt|gap>D @gapinput|y := RandomGroupElement( elms );;D @gapprompt|gap>D @gapinput|z := ImageElm( isom, y );;D @gapprompt|gap>D @gapinput|y = PreImagesRepresentative( isom, z );D true # @gapprompt|gap>D @gapinput|FactorsPolynomialAlgExt( F, g );D [ x_1+(-a), x_1+(a-2), x_1+(-1/7*a^3+3/7*a^2+31/7*a-40/7), x_1+(1/7*a^3-3/7*a^2-31/7*a+26/7) ] \end{Verbatim} } } \chapter{\textcolor{Chapter }{Installation}}\label{Installation} \logpage{[ 4, 0, 0 ]} \hyperdef{L}{X8360C04082558A12}{} { This package provides an interface between \textsf{GAP} and either PARI/GP or OSCAR. When \textsf{Alnuth} is loaded in \textsf{GAP} running inside OSCAR, it automatically uses OSCAR. Otherwise, \textsf{Alnuth} uses PARI/GP, which has to be obtained and installed independently of this package. \textsf{Alnuth} works with PARI/GP Version 2.5 or higher. \section{\textcolor{Chapter }{Installing Alnuth}}\label{Installing Alnuth} \logpage{[ 4, 1, 0 ]} \hyperdef{L}{X821B9C2D817C0BE7}{} { The package \textsf{Alnuth} is part of the standard distribution of \textsf{GAP} so that in most cases there is no need to install it separately. If you are using \textsf{GAP} inside OSCAR, then \textsf{Alnuth} is already bundled and automatically uses OSCAR, so no further installation is required. Otherwise, to use \textsf{Alnuth} you need to have PARI/GP installed. See the following section for information on PARI/GP. In case you want to update \textsf{Alnuth} independently of your main \textsf{GAP} installation or if you are interested in an old version of \textsf{Alnuth} interfacing to KANT/KASH you can find all released versions of \textsf{Alnuth} in the form of gzipped tar\texttt{\symbol{45}}archives at \href{https://github.com/gap-packages/alnuth/releases} {\texttt{https://github.com/gap\texttt{\symbol{45}}packages/alnuth/releases}} There are two ways of installing a \textsf{GAP} package. If you have permission to add files to the installation of \textsf{GAP} on your system you may install \textsf{Alnuth} into the \texttt{pkg} subdirectory of the \textsf{GAP} installation tree. Otherwise you may install \textsf{Alnuth} in a private \texttt{pkg} directory (for details see Subsections (\textbf{Reference: Installing a GAP Package}) and (\textbf{Reference: GAP Root Directories}) in the \textsf{GAP} reference manual). To install the latest version of \textsf{Alnuth} download one of the archives \texttt{alnuth.tar.gz}, move it to the directory \texttt{pkg} in which you want to install, and unpack the archive. If you are using the command line you can unpack with the command \texttt{tar xzf alnuth.tar.gz}. } \section{\textcolor{Chapter }{Getting PARI/GP}}\label{Getting PARI/GP} \logpage{[ 4, 2, 0 ]} \hyperdef{L}{X83B28FDC7FACCE51}{} { Using \textsf{Alnuth} outside OSCAR requires an installation of PARI/GP in Version 2.5 or higher. The software PARI/GP is freely available at \href{https://pari.math.u-bordeaux.fr/} {\texttt{https://pari.math.u\texttt{\symbol{45}}bordeaux.fr/}} Note that the place where PARI/GP is located in your system is independent of the place where \textsf{Alnuth} is installed. \begin{enumerate} \item \emph{Installing under Linux} In many Linux distributions PARI/GP can be installed via the software package manager, but this might sometimes be an older version which cannot be used together with \textsf{Alnuth}. (Starting GP from the command line with the option \texttt{\texttt{\symbol{45}}\texttt{\symbol{45}}version\texttt{\symbol{45}}short} will show you the version number.) If you install PARI/GP from source make sure you install with GMP support for better performance and complete the installation with \texttt{make install} so that you can start GP by just calling \texttt{gp} from the command line. \item \emph{Installing under Windows} For Windows it is sufficient to get the basic GP binary which can be found at \href{https://pari.math.u-bordeaux.fr/download.html} {\texttt{https://pari.math.u\texttt{\symbol{45}}bordeaux.fr/download.html}} \end{enumerate} } \section{\textcolor{Chapter }{Adjust the path of the executable for GP}}\label{Adjust the path of the executable for GP} \logpage{[ 4, 3, 0 ]} \hyperdef{L}{X815DF28A87D65A3A}{} { When \textsf{Alnuth} is used outside OSCAR, it needs to know where the executable for GP is. In the default setting \textsf{Alnuth} looks for an executable program named \texttt{gp} in the search paths of the system. More precisely, for a file \texttt{gp} inside one of the directories in the list returned by \texttt{DirectoriesSystemPrograms()} (called in a \textsf{GAP} session). Under Linux the default setting should work with a standard installation of PARI/GP. For the default setting to work under Windows the downloaded executable file, for example \texttt{gp\texttt{\symbol{45}}2\texttt{\symbol{45}}5\texttt{\symbol{45}}0.exe} has to be renamed to \texttt{gp.exe} and moved to one of the directories listed by \texttt{DirectoriesSystemPrograms()} (Ignore the leading \texttt{cygdrive} in each path name and note that the single letter specifies the drive, for example \texttt{/cygdrive/c/Windows/} denotes the folder \texttt{Windows} on drive \texttt{C:}). To check whether an executable of GP in Version 2.5 or higher is available with the default setting, you can use the function \subsection{\textcolor{Chapter }{PariVersion}} \logpage{[ 4, 3, 1 ]}\nobreak \hyperdef{L}{X84E2E83F8251DC49}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{PariVersion({\mdseries\slshape })\index{PariVersion@\texttt{PariVersion}} \label{PariVersion} }\hfill{\scriptsize (function)}}\\ which prints the version number, if the user preference \texttt{PariGpPath} refers to a usable GP executable. } If you cannot use the default setting for you purpose, you have several ways to configure Alnuth. The recommended approach is to use the \textsf{GAP} user preference system: \begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] !gapprompt@gap>| !gapinput@SetUserPreference("alnuth", "PariGpPath", "/home/my/pari-2.5.0/gp");| !gapprompt@gap>| !gapinput@SetUserPreference("alnuth", "PariStackSize", 256);| !gapprompt@gap>| !gapinput@WriteGapIniFile();| \end{Verbatim} This changes the current session immediately. \texttt{WriteGapIniFile()} writes the current non\texttt{\symbol{45}}default settings to your personal \texttt{gap.ini}, so they are used again in future \textsf{GAP} sessions. See (\textbf{Reference: Configuring User preferences}) for background on the user preference system. The user preferences relevant for Alnuth are shown by \texttt{ShowUserPreferences("alnuth");}. The most important ones are \texttt{PariGpPath}, \texttt{PariGpOptions}, \texttt{PariStackSize}, and \texttt{PrimitiveElementTrials}. In older versions of \textsf{Alnuth} configuration was done via the global variables \texttt{AL{\textunderscore}EXECUTABLE}, \texttt{AL{\textunderscore}PATH}, \texttt{AL{\textunderscore}OPTIONS}, \texttt{AL{\textunderscore}STACKSIZE}, and \texttt{PRIM{\textunderscore}TEST}. These globals are deprecated, and they trigger a warning when used. The globals except for \texttt{AL{\textunderscore}PATH} still serve as a fallback if the corresponding user preference is unusable. Their replacements are as follows: \begin{description} \item[{\texttt{AL{\textunderscore}EXECUTABLE}}] use the user preference \texttt{PariGpPath} \item[{\texttt{AL{\textunderscore}OPTIONS}}] use the user preference \texttt{PariGpOptions} \item[{\texttt{AL{\textunderscore}STACKSIZE}}] use the user preference \texttt{PariStackSize} \item[{\texttt{PRIM{\textunderscore}TEST}}] use the user preference \texttt{PrimitiveElementTrials} \item[{\texttt{AL{\textunderscore}PATH}}] has been removed; Alnuth now always uses the bundled GP helper files. \end{description} For backwards compatibility, the following functions still exist after loading the package (see Section \ref{Loading and testing the package}), but they are deprecated wrappers around the user preference system. \subsection{\textcolor{Chapter }{SetAlnuthExternalExecutable}} \logpage{[ 4, 3, 2 ]}\nobreak \hyperdef{L}{X7F2835FB7C5069DD}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{SetAlnuthExternalExecutable({\mdseries\slshape path})\index{SetAlnuthExternalExecutable@\texttt{SetAlnuthExternalExecutable}} \label{SetAlnuthExternalExecutable} }\hfill{\scriptsize (function)}}\\ sets the user preference \texttt{PariGpPath} for the current \textsf{GAP} session. Depending on your installation of PARI/GP and your operating system the string \mbox{\texttt{\mdseries\slshape path}} can be either the command to start GP in a terminal (for example \texttt{gp}) or the complete path to the executable of GP. The function is deprecated; new code should call \texttt{SetUserPreference("alnuth", "PariGpPath", path);} directly. The function returns the stored preference value. } \subsection{\textcolor{Chapter }{SetAlnuthExternalExecutablePermanently}} \logpage{[ 4, 3, 3 ]}\nobreak \hyperdef{L}{X81CD0C70812FC6F3}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{SetAlnuthExternalExecutablePermanently({\mdseries\slshape path})\index{SetAlnuthExternalExecutablePermanently@\texttt{Set}\-\texttt{Alnuth}\-\texttt{External}\-\texttt{Executable}\-\texttt{Permanently}} \label{SetAlnuthExternalExecutablePermanently} }\hfill{\scriptsize (function)}}\\ does the same as \texttt{SetAlnuthExternalExecutable} and then calls \texttt{WriteGapIniFile()} to persist the current user preference settings in your personal \texttt{gap.ini}. The function is deprecated; new code should call \texttt{SetUserPreference("alnuth", "PariGpPath", path); WriteGapIniFile();} instead. } \subsection{\textcolor{Chapter }{RestoreAlnuthExternalExecutablePermanently}} \logpage{[ 4, 3, 4 ]}\nobreak \hyperdef{L}{X7F648B427EB19552}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{RestoreAlnuthExternalExecutablePermanently({\mdseries\slshape })\index{RestoreAlnuthExternalExecutablePermanently@\texttt{Restore}\-\texttt{Alnuth}\-\texttt{External}\-\texttt{Executable}\-\texttt{Permanently}} \label{RestoreAlnuthExternalExecutablePermanently} }\hfill{\scriptsize (function)}}\\ restores the default value of \texttt{PariGpPath} and then calls \texttt{WriteGapIniFile()}. The function is deprecated; new code should set \texttt{PariGpPath} to the desired default command or path and then call \texttt{WriteGapIniFile()} directly. } } \section{\textcolor{Chapter }{Loading and testing the package}}\label{Loading and testing the package} \logpage{[ 4, 4, 0 ]} \hyperdef{L}{X781C2F107F76F8A4}{} { If \textsf{Alnuth} is not loaded when GAP is started you have to request it explicitly to use it. This is done by calling \texttt{LoadPackage("alnuth");} in a GAP session. If \textsf{Alnuth} had not been loaded already a short banner will be displayed. \begin{Verbatim}[commandchars=!|F,fontsize=\small,frame=single,label=Example] !gapprompt|gap>F !gapinput|LoadPackage("alnuth");F Loading Alnuth 4.0.0 (ALgebraic NUmber THeory and an interface to PARI/GP and OSCAR) by Björn Assmann, Andreas Distler (a.distler@tu-bs.de), and Bettina Eick (http://www.iaa.tu-bs.de/beick). maintained by: Max Horn (https://www.quendi.de/math) and The GAP Team (support@gap-system.org). Homepage: https://gap-packages.github.io/alnuth Report issues at https://github.com/gap-packages/alnuth/issues true gap> \end{Verbatim} To load a certain version of \textsf{Alnuth} you can specify the version number as second argument in the call to \texttt{LoadPackage}. (See \texttt{LoadPackage} (\textbf{Reference: LoadPackage}) in the \textsf{GAP} reference manual or type \texttt{?LoadPackage} within a GAP session). Once the package is loaded, it is possible to check the correct installation running a short test by calling \texttt{ReadPackage("Alnuth", "tst/testinstall.tst");}. \begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] !gapprompt@gap>| !gapinput@ReadPackage("Alnuth", "tst/testinstall.g");| Architecture: aarch64-apple-darwin21.4.0-default64-kv8 testing: GAPROOT/pkg/alnuth/tst/ALNUTH.tst 66 ms (33 ms GC) and 11.0MB allocated for GAPROOT/pkg/alnuth/tst/ALNUTH.tst testing: GAPROOT/pkg/alnuth/tst/version.tst 21 ms (21 ms GC) and 29.6KB allocated for GAPROOT/pkg/alnuth/tst/version.tst ----------------------------------- total 87 ms (54 ms GC) and 11.0MB allocated 0 failures in 2 files #I No errors detected while testing gap> \end{Verbatim} The architecture, timings and memory usage will usually differ; other discrepancies in the output indicate some problem. If the test suite runs into an error in the first part, which when running outside OSCAR verifies the availability of PARI/GP, check your installation of PARI/GP and consult the last chapter of the documentation of \textsf{Alnuth} for more information. If you find any bugs or have any suggestions or comments, we would very much appreciate it if you would let us know by submitting an issue at the \textsf{Alnuth} issue tracker on GitHub \href{https://github.com/gap-packages/alnuth/issues} {\texttt{https://github.com/gap\texttt{\symbol{45}}packages/alnuth/issues}} or by writing an mail to \href{mailto://support@gap-system.org} {\texttt{support@gap\texttt{\symbol{45}}system.org}}. } } {\nobreakspace} \def\bibname{References\logpage{[ "Bib", 0, 0 ]} \hyperdef{L}{X7A6F98FD85F02BFE}{} } \bibliographystyle{alpha} \bibliography{manual.bib} \addcontentsline{toc}{chapter}{References} \def\indexname{Index\logpage{[ "Ind", 0, 0 ]} \hyperdef{L}{X83A0356F839C696F}{} } \cleardoublepage \phantomsection \addcontentsline{toc}{chapter}{Index} \printindex \immediate\write\pagenrlog{["Ind", 0, 0], \arabic{page},} \newpage \immediate\write\pagenrlog{["End"], \arabic{page}];} \immediate\closeout\pagenrlog \end{document} alnuth-4.0.0/doc/_main.xml0000644000000000000000000000073715201210200012274 0ustar00 ] > <#Include SYSTEM "title.xml"> <#Include SYSTEM "intro.xml"> <#Include SYSTEM "fields.xml"> <#Include SYSTEM "example.xml"> <#Include SYSTEM "install.xml"> <#Include SYSTEM "_AutoDocMainFile.xml"> alnuth-4.0.0/doc/chap0.html0000644000000000000000000002517315201210200012351 0ustar00 GAP (Alnuth) - Contents
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Alnuth

ALgebraic NUmber THeory and an interface to PARI/GP and OSCAR

4.0.0

14 May 2026

Note: PARI/GP is not part of this package. It can be obtained from https://pari.math.u-bordeaux.fr/. If you use GAP via OSCAR, then OSCAR will automatically be used instead of PARI/GP.

GAP code by Björn Assmann, Andreas Distler and Bettina Eick

PARI/GP code by Bill Allombert

OSCAR code by Claus Fieker and Max Horn

Copyright

This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 2 of the license, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this program. If not, see https://www.gnu.org/licenses/

Acknowledgements

To begin with we are very grateful for all the feedback by users of former versions of Alnuth.

We thank Bill Allombert who wrote the GP code for the interface to PARI/GP and who was extremely helpful in the transition from KANT to PARI/GP.

For feedback on the development version, including a code patch, we are much obliged to Max Horn.

The second author acknowledges the financial support at CAUL within the projects PTDC/MAT/101993/2008 and ISFL-1-143, financed by FEDER and FCT, in the development of Version 3 of Alnuth.

Support for using OSCAR instead of PARI/GP was added by Claus Fieker and Max Horn.

Contents

1 Introduction
2 Working with number fields
 2.1 Creation of number fields

  2.1-1 FieldByMatricesNC
  2.1-2 FieldByMatrixBasisNC
  2.1-3 FieldByPolynomialNC
 2.2 Methods for number fields

  2.2-1 PrimitiveElement
  2.2-2 IsPrimitiveElementOfNumberField
  2.2-3 DegreeOverPrimeField
  2.2-4 EquationOrderBasis
  2.2-5 UnitGroup
  2.2-6 IsUnitOfNumberField
  2.2-7 ExponentsOfUnits
  2.2-8 IsCyclotomicField
  2.2-9 NormCosetsOfNumberField
 2.3 Presentations of multiplicative subgroups

  2.3-1 PcpPresentationOfMultiplicativeSubgroup
  2.3-2 RelationLattice
 2.4 Methods to compute with subgroups of the unit group

  2.4-1 RelationLatticeOfUnits
  2.4-2 IntersectionOfUnitSubgroups
 2.5 Factorisation of polynomials over a number field

  2.5-1 FactorsPolynomialAlgExt
  2.5-2 FactorsPolynomialAlnuth
 2.6 Examples

  2.6-1 ExampleMatField
3 An example application
 3.1 Number fields defined by matrices
 3.2 Number fields defined by a polynomial
4 Installation
 4.1 Installing Alnuth
 4.2 Getting PARI/GP
 4.3 Adjust the path of the executable for GP

  4.3-1 PariVersion
  4.3-2 SetAlnuthExternalExecutable
  4.3-3 SetAlnuthExternalExecutablePermanently
  4.3-4 RestoreAlnuthExternalExecutablePermanently
 4.4 Loading and testing the package
References
Index

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alnuth-4.0.0/doc/chap0.txt0000644000000000000000000001011315201210200012210 0ustar00  Alnuth   ALgebraic NUmber THeory and an interface to PARI/GP and OSCAR  4.0.0 14 May 2026 GAP code by Björn Assmann, Andreas Distler and Bettina Eick PARI/GP code by Bill Allombert OSCAR code by Claus Fieker and Max Horn Note: PARI/GP is not part of this package. It can be obtained from https://pari.math.u-bordeaux.fr/. If you use GAP via OSCAR, then OSCAR will automatically be used instead of PARI/GP. ------------------------------------------------------- Copyright This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 2 of the license, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program. If not, see https://www.gnu.org/licenses/ ------------------------------------------------------- Acknowledgements To begin with we are very grateful for all the feedback by users of former versions of Alnuth. We thank Bill Allombert who wrote the GP code for the interface to PARI/GP and who was extremely helpful in the transition from KANT to PARI/GP. For feedback on the development version, including a code patch, we are much obliged to Max Horn. The second author acknowledges the financial support at CAUL within the projects PTDC/MAT/101993/2008 and ISFL-1-143, financed by FEDER and FCT, in the development of Version 3 of Alnuth. Support for using OSCAR instead of PARI/GP was added by Claus Fieker and Max Horn. ------------------------------------------------------- Contents (Alnuth) 1 Introduction 2 Working with number fields 2.1 Creation of number fields 2.1-1 FieldByMatricesNC 2.1-2 FieldByMatrixBasisNC 2.1-3 FieldByPolynomialNC 2.2 Methods for number fields 2.2-1 PrimitiveElement 2.2-2 IsPrimitiveElementOfNumberField 2.2-3 DegreeOverPrimeField 2.2-4 EquationOrderBasis 2.2-5 UnitGroup 2.2-6 IsUnitOfNumberField 2.2-7 ExponentsOfUnits 2.2-8 IsCyclotomicField 2.2-9 NormCosetsOfNumberField 2.3 Presentations of multiplicative subgroups 2.3-1 PcpPresentationOfMultiplicativeSubgroup 2.3-2 RelationLattice 2.4 Methods to compute with subgroups of the unit group 2.4-1 RelationLatticeOfUnits 2.4-2 IntersectionOfUnitSubgroups 2.5 Factorisation of polynomials over a number field 2.5-1 FactorsPolynomialAlgExt 2.5-2 FactorsPolynomialAlnuth 2.6 Examples 2.6-1 ExampleMatField 3 An example application 3.1 Number fields defined by matrices 3.2 Number fields defined by a polynomial 4 Installation 4.1 Installing Alnuth 4.2 Getting PARI/GP 4.3 Adjust the path of the executable for GP 4.3-1 PariVersion 4.3-2 SetAlnuthExternalExecutable 4.3-3 SetAlnuthExternalExecutablePermanently 4.3-4 RestoreAlnuthExternalExecutablePermanently 4.4 Loading and testing the package  alnuth-4.0.0/doc/chap0_mj.html0000644000000000000000000002566115201210200013041 0ustar00 GAP (Alnuth) - Contents
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Alnuth

ALgebraic NUmber THeory and an interface to PARI/GP and OSCAR

4.0.0

14 May 2026

Note: PARI/GP is not part of this package. It can be obtained from https://pari.math.u-bordeaux.fr/. If you use GAP via OSCAR, then OSCAR will automatically be used instead of PARI/GP.

GAP code by Björn Assmann, Andreas Distler and Bettina Eick

PARI/GP code by Bill Allombert

OSCAR code by Claus Fieker and Max Horn

Copyright

This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 2 of the license, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this program. If not, see https://www.gnu.org/licenses/

Acknowledgements

To begin with we are very grateful for all the feedback by users of former versions of Alnuth.

We thank Bill Allombert who wrote the GP code for the interface to PARI/GP and who was extremely helpful in the transition from KANT to PARI/GP.

For feedback on the development version, including a code patch, we are much obliged to Max Horn.

The second author acknowledges the financial support at CAUL within the projects PTDC/MAT/101993/2008 and ISFL-1-143, financed by FEDER and FCT, in the development of Version 3 of Alnuth.

Support for using OSCAR instead of PARI/GP was added by Claus Fieker and Max Horn.

Contents

1 Introduction
2 Working with number fields
 2.1 Creation of number fields

  2.1-1 FieldByMatricesNC
  2.1-2 FieldByMatrixBasisNC
  2.1-3 FieldByPolynomialNC
 2.2 Methods for number fields

  2.2-1 PrimitiveElement
  2.2-2 IsPrimitiveElementOfNumberField
  2.2-3 DegreeOverPrimeField
  2.2-4 EquationOrderBasis
  2.2-5 UnitGroup
  2.2-6 IsUnitOfNumberField
  2.2-7 ExponentsOfUnits
  2.2-8 IsCyclotomicField
  2.2-9 NormCosetsOfNumberField
 2.3 Presentations of multiplicative subgroups

  2.3-1 PcpPresentationOfMultiplicativeSubgroup
  2.3-2 RelationLattice
 2.4 Methods to compute with subgroups of the unit group

  2.4-1 RelationLatticeOfUnits
  2.4-2 IntersectionOfUnitSubgroups
 2.5 Factorisation of polynomials over a number field

  2.5-1 FactorsPolynomialAlgExt
  2.5-2 FactorsPolynomialAlnuth
 2.6 Examples

  2.6-1 ExampleMatField
3 An example application
 3.1 Number fields defined by matrices
 3.2 Number fields defined by a polynomial
4 Installation
 4.1 Installing Alnuth
 4.2 Getting PARI/GP
 4.3 Adjust the path of the executable for GP

  4.3-1 PariVersion
  4.3-2 SetAlnuthExternalExecutable
  4.3-3 SetAlnuthExternalExecutablePermanently
  4.3-4 RestoreAlnuthExternalExecutablePermanently
 4.4 Loading and testing the package
References
Index

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1 Introduction

1 Introduction

A number field is a finite extension of the field of rational numbers. Alnuth provides various methods to compute with number fields which are given by a defining polynomial or by generators. For background on number fields we refer to [ST79].

Some of the methods provided in this package are written in GAP code. The other part of the methods is imported from the Computer Algebra Systems PARI/GP [PAR11] respectively OSCAR [DEF+25], [OSC24]. Hence this package contains some GAP functions and an interface to some functions to these computer algebra systems. Therefore one has to have PARI/GP or OSCAR installed to use the full functionality of Alnuth.

We note that only a very small part of the functions available in PARI/GP respectively OSCAR are linked to GAP and they provides many more methods for computations in number fields.

The main methods included in Alnuth are: creating a number field, computing its maximal order, computing its unit group and a presentation of this unit group, computing the elements of a given norm of the number field, determining a presentation for a finitely generated multiplicative subgroup, and factoring polynomials defined over number fields. For background on algorithms for number fields we refer to [Poh93], [PZ89] and [Coh93].

The functions provided by Alnuth are introduced in the following chapter. Then an example application is outlined. In the final chapter of this manual the installation of the package and configuration of the interface, including hints on the installation of PARI/GP respectively OSCAR, are described.

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alnuth-4.0.0/doc/chap1.txt0000644000000000000000000000360115201210200012215 0ustar00 1 Introduction A number field is a finite extension of the field of rational numbers. Alnuth provides various methods to compute with number fields which are given by a defining polynomial or by generators. For background on number fields we refer to [ST79]. Some of the methods provided in this package are written in GAP code. The other part of the methods is imported from the Computer Algebra Systems PARI/GP [PAR11] respectively OSCAR [DEF+25], [OSC24]. Hence this package contains some GAP functions and an interface to some functions to these computer algebra systems. Therefore one has to have PARI/GP or OSCAR installed to use the full functionality of Alnuth. We note that only a very small part of the functions available in PARI/GP respectively OSCAR are linked to GAP and they provides many more methods for computations in number fields. The main methods included in Alnuth are: creating a number field, computing its maximal order, computing its unit group and a presentation of this unit group, computing the elements of a given norm of the number field, determining a presentation for a finitely generated multiplicative subgroup, and factoring polynomials defined over number fields. For background on algorithms for number fields we refer to [Poh93], [PZ89] and [Coh93]. The functions provided by Alnuth are introduced in the following chapter. Then an example application is outlined. In the final chapter of this manual the installation of the package and configuration of the interface, including hints on the installation of PARI/GP respectively OSCAR, are described. alnuth-4.0.0/doc/chap1_mj.html0000644000000000000000000001052315201210200013031 0ustar00 GAP (Alnuth) - Chapter 1: Introduction
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1 Introduction

1 Introduction

A number field is a finite extension of the field of rational numbers. Alnuth provides various methods to compute with number fields which are given by a defining polynomial or by generators. For background on number fields we refer to [ST79].

Some of the methods provided in this package are written in GAP code. The other part of the methods is imported from the Computer Algebra Systems PARI/GP [PAR11] respectively OSCAR [DEF+25], [OSC24]. Hence this package contains some GAP functions and an interface to some functions to these computer algebra systems. Therefore one has to have PARI/GP or OSCAR installed to use the full functionality of Alnuth.

We note that only a very small part of the functions available in PARI/GP respectively OSCAR are linked to GAP and they provides many more methods for computations in number fields.

The main methods included in Alnuth are: creating a number field, computing its maximal order, computing its unit group and a presentation of this unit group, computing the elements of a given norm of the number field, determining a presentation for a finitely generated multiplicative subgroup, and factoring polynomials defined over number fields. For background on algorithms for number fields we refer to [Poh93], [PZ89] and [Coh93].

The functions provided by Alnuth are introduced in the following chapter. Then an example application is outlined. In the final chapter of this manual the installation of the package and configuration of the interface, including hints on the installation of PARI/GP respectively OSCAR, are described.

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2 Working with number fields
 2.1 Creation of number fields

  2.1-1 FieldByMatricesNC
  2.1-2 FieldByMatrixBasisNC
  2.1-3 FieldByPolynomialNC
 2.2 Methods for number fields

  2.2-1 PrimitiveElement
  2.2-2 IsPrimitiveElementOfNumberField
  2.2-3 DegreeOverPrimeField
  2.2-4 EquationOrderBasis
  2.2-5 UnitGroup
  2.2-6 IsUnitOfNumberField
  2.2-7 ExponentsOfUnits
  2.2-8 IsCyclotomicField
  2.2-9 NormCosetsOfNumberField
 2.3 Presentations of multiplicative subgroups

  2.3-1 PcpPresentationOfMultiplicativeSubgroup
  2.3-2 RelationLattice
 2.4 Methods to compute with subgroups of the unit group

  2.4-1 RelationLatticeOfUnits
  2.4-2 IntersectionOfUnitSubgroups
 2.5 Factorisation of polynomials over a number field

  2.5-1 FactorsPolynomialAlgExt
  2.5-2 FactorsPolynomialAlnuth
 2.6 Examples

  2.6-1 ExampleMatField

2 Working with number fields

An algebraic number field is a finite-dimensional extension of the rational numbers ℚ. Such a number field has a primitive element and it can be defined by the minimal polynomial of this primitive element. Another important way to define an algebraic number field is by a set of rational matrices which generate a number field.

2.1 Creation of number fields

We provide functions to create number fields defined by rational matrices or by rational polynomials.

2.1-1 FieldByMatricesNC
‣ FieldByMatricesNC( matrices )( function )
‣ FieldByMatrices( matrices )( function )

Creates a field generated by the rational matrices matrices. In the faster NC version, the function assumes that the input generates a field and there are no checks on this performed.

2.1-2 FieldByMatrixBasisNC
‣ FieldByMatrixBasisNC( matrices )( function )
‣ FieldByMatrixBasis( matrices )( function )

Creates a field with basis matrices. The list matrices must consist of rational matrices which form a basis for a number field. In the faster NC version, the function assumes that the input is a matrix basis for a field and no checks are performed.

2.1-3 FieldByPolynomialNC
‣ FieldByPolynomialNC( polynomial )( function )
‣ FieldByPolynomial( polynomial )( function )

Creates a field defined by polynomial. The polynomial polynomial must be an irreducible rational polynomial. In the faster NC version, no checks on the input are performed.

2.2 Methods for number fields

We outline a number of functions for number fields.

2.2-1 PrimitiveElement
‣ PrimitiveElement( F )( function )
‣ DefiningPolynomial( F )( function )

Computes a primitive element and a defining polynomial for the given number field. The defining polynomial is the minimal polynomial of the primitive element. Since F contains various primitive elements, PrimitiveElement tries to find a primitive element which has a minimal polynomial with small coefficients. Via the user preference PrimitiveElementTrials the user can decide how many primitive elements will be compared. The default value is 20. (See SetUserPreference (Reference: SetUserPreference) for more information about user preferences.)

2.2-2 IsPrimitiveElementOfNumberField
‣ IsPrimitiveElementOfNumberField( F, a )( function )

Checks if the given element generates the field.

2.2-3 DegreeOverPrimeField
‣ DegreeOverPrimeField( F )( function )

Returns the degree of F over the rationals.

2.2-4 EquationOrderBasis
‣ EquationOrderBasis( F )( function )
‣ MaximalOrderBasis( F )( function )
‣ IsIntegerOfNumberField( F, k )( function )

These functions return bases for the equation order or the maximal order of the number field F. Also, they allow to check if a given element is an integer in the given number field.

2.2-5 UnitGroup
‣ UnitGroup( F )( function )

determines the unit group of F.

Recall that the unit group of F is a finitely generated abelian group. The function IsomorphismPcpGroup from the Polycyclic [EHN11] package gives an isomorphism to a pcp group which can be used for various computations with the unit group.

2.2-6 IsUnitOfNumberField
‣ IsUnitOfNumberField( F, k )( function )

checks whether the element k is a unit in F.

2.2-7 ExponentsOfUnits
‣ ExponentsOfUnits( F, elms )( function )

This function determines the exponent vectors of the elements in elms with respect to the generators of the unit group of F. If the unit group of F is not known, then the function computes this unit group also.

2.2-8 IsCyclotomicField
‣ IsCyclotomicField( F )( function )

Check whether F is cyclotomic.

2.2-9 NormCosetsOfNumberField
‣ NormCosetsOfNumberField( F, norm )( function )

Returns a description for the set of all elements of norm norm in F. These elements can be written as a finite union of cosets of the unit group of F. The function returns coset representatives for these cosets.

2.3 Presentations of multiplicative subgroups

Suppose that a finite number of invertible elements of a number field are given. Then these elements generate a finitely generated abelian group. However, it is a non-trivial task to provide a presentation for this abelian group. The most useful representation for such groups is as pcp group.

2.3-1 PcpPresentationOfMultiplicativeSubgroup
‣ PcpPresentationOfMultiplicativeSubgroup( F, elms )( function )
‣ IsomorphismPcpGroup( F, elms )( function )

Determine a pcp presentation for the multiplicative group of Fbackslash{0} generated by elms and an isomorphism on this presentation. Note, that the method IsomorphismPcpGroup is defined in the Polycyclic package [EHN11]. We refer to the manual of this package for further background.

In the determination of the Pcp-presentation of a multiplicative subgroup generated by elms the relations between the elements in elms play an important role. Let elms={e_1,dots,e_l} be a finite subset of a field F. The relation lattice for elms is

rl(elms):=\left\{(h_1,\dots,h_l) \in ℤ^l | e_1^{h_1} \cdots e_l^{h_l} = 1\right\} .

2.3-2 RelationLattice
‣ RelationLattice( F, elms )( function )

Determines a generating set for the relation lattice of the field elements elms.

2.4 Methods to compute with subgroups of the unit group

2.4-1 RelationLatticeOfUnits
‣ RelationLatticeOfUnits( F, elms )( function )

Determines a basis for the relation lattice of the units elms in triangularized form. Note that this method is more efficient than the method RelationLattice.

2.4-2 IntersectionOfUnitSubgroups
‣ IntersectionOfUnitSubgroups( F, gen1, gen2 )( function )

The lists gen1 and gen2 are supposed to generate two subgroups U_1 and U_2 of the unit group of F. This function determines the intersection of U_1 with U_2. The result is returned as a list of vectors generating the lattice { e ∈ ℤ^n ∣ g_1^e_1 ⋯ g_n^e_n ∈ U_2 } for gen1 = [g_1, ..., g_n].

For efficiency reasons this function does not check the input and it may return wrong results if the input generators do not fulfil the requirements.

2.5 Factorisation of polynomials over a number field

2.5-1 FactorsPolynomialAlgExt
‣ FactorsPolynomialAlgExt( F, pol )( function )

embeds the rational polynomial pol into the polynomial ring over the number field F, which has to be constructed by FieldByPolynomial or AlgebraicExtension, and returns the factorization of the embedded polynomial. By default a denotes the primitive element of the field one can obtain from PrimitiveElement(F), that is, a root of the defining polynomial of F.

2.5-2 FactorsPolynomialAlnuth
‣ FactorsPolynomialAlnuth( pol )( function )

takes a polynomial pol defined over an algebraic extension of the Rationals and factors it using the external CAS (PARI/GP or OSCAR).

The alias FactorsPolynomialPari is provided for backwards compatibility.

gap> x := Indeterminate( Rationals, "x" );;
gap> pol := 2*x^7+2*x^5+8*x^4+8*x^2;
2*x^7+2*x^5+8*x^4+8*x^2
gap> L := FieldByPolynomial( x^3-4 );
<algebraic extension over the Rationals of degree 3>
gap> y := Indeterminate( L, "y" );;
gap> FactorsPolynomialAlgExt( L, pol );
[ !2*y, y, y+a, y^2+!1, y^2+(-a)*y+a^2 ]
gap> FactorsPolynomialAlnuth( last[5] );
[ y^2+(-a)*y+a^2 ]

2.6 Examples

2.6-1 ExampleMatField
‣ ExampleMatField( l )( function )

This function returns some examples of fields generated by matrices. There are 9 such example fields provided and they can be obtained by assigning the input l to an integer between 1 and 9. Some of the properties of the examples are summarized in the following table.

                    degree over Q  number of generators  dim. of generators
ExampleMatField(1)              4                     4                   4
ExampleMatField(2)              4                     4                   4
ExampleMatField(3)              4                     4                   4
ExampleMatField(4)              4                    13                   4
ExampleMatField(5)              4                    13                   4
ExampleMatField(6)              4                     7                   4
ExampleMatField(7)              4                    18                   4
ExampleMatField(8)              4                    13                   4
ExampleMatField(9)              4                     7                   4
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alnuth-4.0.0/doc/chap2.txt0000644000000000000000000002775515201210200012236 0ustar00 2 Working with number fields An algebraic number field is a finite-dimensional extension of the rational numbers ℚ. Such a number field has a primitive element and it can be defined by the minimal polynomial of this primitive element. Another important way to define an algebraic number field is by a set of rational matrices which generate a number field. 2.1 Creation of number fields We provide functions to create number fields defined by rational matrices or by rational polynomials. 2.1-1 FieldByMatricesNC FieldByMatricesNC( matrices )  function FieldByMatrices( matrices )  function Creates a field generated by the rational matrices matrices. In the faster NC version, the function assumes that the input generates a field and there are no checks on this performed. 2.1-2 FieldByMatrixBasisNC FieldByMatrixBasisNC( matrices )  function FieldByMatrixBasis( matrices )  function Creates a field with basis matrices. The list matrices must consist of rational matrices which form a basis for a number field. In the faster NC version, the function assumes that the input is a matrix basis for a field and no checks are performed. 2.1-3 FieldByPolynomialNC FieldByPolynomialNC( polynomial )  function FieldByPolynomial( polynomial )  function Creates a field defined by polynomial. The polynomial polynomial must be an irreducible rational polynomial. In the faster NC version, no checks on the input are performed. 2.2 Methods for number fields We outline a number of functions for number fields. 2.2-1 PrimitiveElement PrimitiveElement( F )  function DefiningPolynomial( F )  function Computes a primitive element and a defining polynomial for the given number field. The defining polynomial is the minimal polynomial of the primitive element. Since F contains various primitive elements, PrimitiveElement tries to find a primitive element which has a minimal polynomial with small coefficients. Via the user preference PrimitiveElementTrials the user can decide how many primitive elements will be compared. The default value is 20. (See SetUserPreference (Reference: SetUserPreference) for more information about user preferences.) 2.2-2 IsPrimitiveElementOfNumberField IsPrimitiveElementOfNumberField( F, a )  function Checks if the given element generates the field. 2.2-3 DegreeOverPrimeField DegreeOverPrimeField( F )  function Returns the degree of F over the rationals. 2.2-4 EquationOrderBasis EquationOrderBasis( F )  function MaximalOrderBasis( F )  function IsIntegerOfNumberField( F, k )  function These functions return bases for the equation order or the maximal order of the number field F. Also, they allow to check if a given element is an integer in the given number field. 2.2-5 UnitGroup UnitGroup( F )  function determines the unit group of F. Recall that the unit group of F is a finitely generated abelian group. The function IsomorphismPcpGroup from the Polycyclic [EHN11] package gives an isomorphism to a pcp group which can be used for various computations with the unit group. 2.2-6 IsUnitOfNumberField IsUnitOfNumberField( F, k )  function checks whether the element k is a unit in F. 2.2-7 ExponentsOfUnits ExponentsOfUnits( F, elms )  function This function determines the exponent vectors of the elements in elms with respect to the generators of the unit group of F. If the unit group of F is not known, then the function computes this unit group also. 2.2-8 IsCyclotomicField IsCyclotomicField( F )  function Check whether F is cyclotomic. 2.2-9 NormCosetsOfNumberField NormCosetsOfNumberField( F, norm )  function Returns a description for the set of all elements of norm norm in F. These elements can be written as a finite union of cosets of the unit group of F. The function returns coset representatives for these cosets. 2.3 Presentations of multiplicative subgroups Suppose that a finite number of invertible elements of a number field are given. Then these elements generate a finitely generated abelian group. However, it is a non-trivial task to provide a presentation for this abelian group. The most useful representation for such groups is as pcp group. 2.3-1 PcpPresentationOfMultiplicativeSubgroup PcpPresentationOfMultiplicativeSubgroup( F, elms )  function IsomorphismPcpGroup( F, elms )  function Determine a pcp presentation for the multiplicative group of Fbackslash{0} generated by elms and an isomorphism on this presentation. Note, that the method IsomorphismPcpGroup is defined in the Polycyclic package [EHN11]. We refer to the manual of this package for further background. In the determination of the Pcp-presentation of a multiplicative subgroup generated by elms the relations between the elements in elms play an important role. Let elms={e_1,dots,e_l} be a finite subset of a field F. The relation lattice for elms is rl(elms):=\left\{(h_1,\dots,h_l) \in ℤ^l | e_1^{h_1} \cdots e_l^{h_l} = 1\right\} .  2.3-2 RelationLattice RelationLattice( F, elms )  function Determines a generating set for the relation lattice of the field elements elms. 2.4 Methods to compute with subgroups of the unit group 2.4-1 RelationLatticeOfUnits RelationLatticeOfUnits( F, elms )  function Determines a basis for the relation lattice of the units elms in triangularized form. Note that this method is more efficient than the method RelationLattice. 2.4-2 IntersectionOfUnitSubgroups IntersectionOfUnitSubgroups( F, gen1, gen2 )  function The lists gen1 and gen2 are supposed to generate two subgroups U_1 and U_2 of the unit group of F. This function determines the intersection of U_1 with U_2. The result is returned as a list of vectors generating the lattice { e ∈ ℤ^n ∣ g_1^e_1 ⋯ g_n^e_n ∈ U_2 } for gen1 = [g_1, ..., g_n]. For efficiency reasons this function does not check the input and it may return wrong results if the input generators do not fulfil the requirements. 2.5 Factorisation of polynomials over a number field 2.5-1 FactorsPolynomialAlgExt FactorsPolynomialAlgExt( F, pol )  function embeds the rational polynomial pol into the polynomial ring over the number field F, which has to be constructed by FieldByPolynomial or AlgebraicExtension, and returns the factorization of the embedded polynomial. By default a denotes the primitive element of the field one can obtain from PrimitiveElement(F), that is, a root of the defining polynomial of F. 2.5-2 FactorsPolynomialAlnuth FactorsPolynomialAlnuth( pol )  function takes a polynomial pol defined over an algebraic extension of the Rationals and factors it using the external CAS (PARI/GP or OSCAR). The alias FactorsPolynomialPari is provided for backwards compatibility.  Example  gap> x := Indeterminate( Rationals, "x" );; gap> pol := 2*x^7+2*x^5+8*x^4+8*x^2; 2*x^7+2*x^5+8*x^4+8*x^2 gap> L := FieldByPolynomial( x^3-4 );  gap> y := Indeterminate( L, "y" );; gap> FactorsPolynomialAlgExt( L, pol ); [ !2*y, y, y+a, y^2+!1, y^2+(-a)*y+a^2 ] gap> FactorsPolynomialAlnuth( last[5] ); [ y^2+(-a)*y+a^2 ]  2.6 Examples 2.6-1 ExampleMatField ExampleMatField( l )  function This function returns some examples of fields generated by matrices. There are 9 such example fields provided and they can be obtained by assigning the input l to an integer between 1 and 9. Some of the properties of the examples are summarized in the following table.  Example   degree over Q number of generators dim. of generators ExampleMatField(1) 4 4 4 ExampleMatField(2) 4 4 4 ExampleMatField(3) 4 4 4 ExampleMatField(4) 4 13 4 ExampleMatField(5) 4 13 4 ExampleMatField(6) 4 7 4 ExampleMatField(7) 4 18 4 ExampleMatField(8) 4 13 4 ExampleMatField(9) 4 7 4  alnuth-4.0.0/doc/chap2_mj.html0000644000000000000000000005743715201210200013051 0ustar00 GAP (Alnuth) - Chapter 2: Working with number fields
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2 Working with number fields
 2.1 Creation of number fields

  2.1-1 FieldByMatricesNC
  2.1-2 FieldByMatrixBasisNC
  2.1-3 FieldByPolynomialNC
 2.2 Methods for number fields

  2.2-1 PrimitiveElement
  2.2-2 IsPrimitiveElementOfNumberField
  2.2-3 DegreeOverPrimeField
  2.2-4 EquationOrderBasis
  2.2-5 UnitGroup
  2.2-6 IsUnitOfNumberField
  2.2-7 ExponentsOfUnits
  2.2-8 IsCyclotomicField
  2.2-9 NormCosetsOfNumberField
 2.3 Presentations of multiplicative subgroups

  2.3-1 PcpPresentationOfMultiplicativeSubgroup
  2.3-2 RelationLattice
 2.4 Methods to compute with subgroups of the unit group

  2.4-1 RelationLatticeOfUnits
  2.4-2 IntersectionOfUnitSubgroups
 2.5 Factorisation of polynomials over a number field

  2.5-1 FactorsPolynomialAlgExt
  2.5-2 FactorsPolynomialAlnuth
 2.6 Examples

  2.6-1 ExampleMatField

2 Working with number fields

An algebraic number field is a finite-dimensional extension of the rational numbers \(ℚ\). Such a number field has a primitive element and it can be defined by the minimal polynomial of this primitive element. Another important way to define an algebraic number field is by a set of rational matrices which generate a number field.

2.1 Creation of number fields

We provide functions to create number fields defined by rational matrices or by rational polynomials.

2.1-1 FieldByMatricesNC
‣ FieldByMatricesNC( matrices )( function )
‣ FieldByMatrices( matrices )( function )

Creates a field generated by the rational matrices matrices. In the faster NC version, the function assumes that the input generates a field and there are no checks on this performed.

2.1-2 FieldByMatrixBasisNC
‣ FieldByMatrixBasisNC( matrices )( function )
‣ FieldByMatrixBasis( matrices )( function )

Creates a field with basis matrices. The list matrices must consist of rational matrices which form a basis for a number field. In the faster NC version, the function assumes that the input is a matrix basis for a field and no checks are performed.

2.1-3 FieldByPolynomialNC
‣ FieldByPolynomialNC( polynomial )( function )
‣ FieldByPolynomial( polynomial )( function )

Creates a field defined by polynomial. The polynomial polynomial must be an irreducible rational polynomial. In the faster NC version, no checks on the input are performed.

2.2 Methods for number fields

We outline a number of functions for number fields.

2.2-1 PrimitiveElement
‣ PrimitiveElement( F )( function )
‣ DefiningPolynomial( F )( function )

Computes a primitive element and a defining polynomial for the given number field. The defining polynomial is the minimal polynomial of the primitive element. Since F contains various primitive elements, PrimitiveElement tries to find a primitive element which has a minimal polynomial with small coefficients. Via the user preference PrimitiveElementTrials the user can decide how many primitive elements will be compared. The default value is 20. (See SetUserPreference (Reference: SetUserPreference) for more information about user preferences.)

2.2-2 IsPrimitiveElementOfNumberField
‣ IsPrimitiveElementOfNumberField( F, a )( function )

Checks if the given element generates the field.

2.2-3 DegreeOverPrimeField
‣ DegreeOverPrimeField( F )( function )

Returns the degree of F over the rationals.

2.2-4 EquationOrderBasis
‣ EquationOrderBasis( F )( function )
‣ MaximalOrderBasis( F )( function )
‣ IsIntegerOfNumberField( F, k )( function )

These functions return bases for the equation order or the maximal order of the number field F. Also, they allow to check if a given element is an integer in the given number field.

2.2-5 UnitGroup
‣ UnitGroup( F )( function )

determines the unit group of F.

Recall that the unit group of F is a finitely generated abelian group. The function IsomorphismPcpGroup from the Polycyclic [EHN11] package gives an isomorphism to a pcp group which can be used for various computations with the unit group.

2.2-6 IsUnitOfNumberField
‣ IsUnitOfNumberField( F, k )( function )

checks whether the element k is a unit in F.

2.2-7 ExponentsOfUnits
‣ ExponentsOfUnits( F, elms )( function )

This function determines the exponent vectors of the elements in elms with respect to the generators of the unit group of F. If the unit group of F is not known, then the function computes this unit group also.

2.2-8 IsCyclotomicField
‣ IsCyclotomicField( F )( function )

Check whether F is cyclotomic.

2.2-9 NormCosetsOfNumberField
‣ NormCosetsOfNumberField( F, norm )( function )

Returns a description for the set of all elements of norm norm in F. These elements can be written as a finite union of cosets of the unit group of F. The function returns coset representatives for these cosets.

2.3 Presentations of multiplicative subgroups

Suppose that a finite number of invertible elements of a number field are given. Then these elements generate a finitely generated abelian group. However, it is a non-trivial task to provide a presentation for this abelian group. The most useful representation for such groups is as pcp group.

2.3-1 PcpPresentationOfMultiplicativeSubgroup
‣ PcpPresentationOfMultiplicativeSubgroup( F, elms )( function )
‣ IsomorphismPcpGroup( F, elms )( function )

Determine a pcp presentation for the multiplicative group of \(\textit{F}\backslash\{0\}\) generated by elms and an isomorphism on this presentation. Note, that the method IsomorphismPcpGroup is defined in the Polycyclic package [EHN11]. We refer to the manual of this package for further background.

In the determination of the Pcp-presentation of a multiplicative subgroup generated by elms the relations between the elements in elms play an important role. Let \(elms=\{e_1,\dots,e_l\}\) be a finite subset of a field F. The relation lattice for elms is

\[ rl(elms):=\left\{(h_1,\dots,h_l) \in ℤ^l | e_1^{h_1} \cdots e_l^{h_l} = 1\right\} . \]

2.3-2 RelationLattice
‣ RelationLattice( F, elms )( function )

Determines a generating set for the relation lattice of the field elements elms.

2.4 Methods to compute with subgroups of the unit group

2.4-1 RelationLatticeOfUnits
‣ RelationLatticeOfUnits( F, elms )( function )

Determines a basis for the relation lattice of the units elms in triangularized form. Note that this method is more efficient than the method RelationLattice.

2.4-2 IntersectionOfUnitSubgroups
‣ IntersectionOfUnitSubgroups( F, gen1, gen2 )( function )

The lists gen1 and gen2 are supposed to generate two subgroups \(U_1\) and \(U_2\) of the unit group of F. This function determines the intersection of \(U_1\) with \(U_2\). The result is returned as a list of vectors generating the lattice \(\{ e \in ℤ^n \mid g_1^{e_1} \cdots g_n^{e_n} \in U_2 \}\) for gen1 = \([g_1, \ldots, g_n]\).

For efficiency reasons this function does not check the input and it may return wrong results if the input generators do not fulfil the requirements.

2.5 Factorisation of polynomials over a number field

2.5-1 FactorsPolynomialAlgExt
‣ FactorsPolynomialAlgExt( F, pol )( function )

embeds the rational polynomial pol into the polynomial ring over the number field F, which has to be constructed by FieldByPolynomial or AlgebraicExtension, and returns the factorization of the embedded polynomial. By default a denotes the primitive element of the field one can obtain from PrimitiveElement(F), that is, a root of the defining polynomial of F.

2.5-2 FactorsPolynomialAlnuth
‣ FactorsPolynomialAlnuth( pol )( function )

takes a polynomial pol defined over an algebraic extension of the Rationals and factors it using the external CAS (PARI/GP or OSCAR).

The alias FactorsPolynomialPari is provided for backwards compatibility.

gap> x := Indeterminate( Rationals, "x" );;
gap> pol := 2*x^7+2*x^5+8*x^4+8*x^2;
2*x^7+2*x^5+8*x^4+8*x^2
gap> L := FieldByPolynomial( x^3-4 );
<algebraic extension over the Rationals of degree 3>
gap> y := Indeterminate( L, "y" );;
gap> FactorsPolynomialAlgExt( L, pol );
[ !2*y, y, y+a, y^2+!1, y^2+(-a)*y+a^2 ]
gap> FactorsPolynomialAlnuth( last[5] );
[ y^2+(-a)*y+a^2 ]

2.6 Examples

2.6-1 ExampleMatField
‣ ExampleMatField( l )( function )

This function returns some examples of fields generated by matrices. There are 9 such example fields provided and they can be obtained by assigning the input l to an integer between 1 and 9. Some of the properties of the examples are summarized in the following table.

                    degree over Q  number of generators  dim. of generators
ExampleMatField(1)              4                     4                   4
ExampleMatField(2)              4                     4                   4
ExampleMatField(3)              4                     4                   4
ExampleMatField(4)              4                    13                   4
ExampleMatField(5)              4                    13                   4
ExampleMatField(6)              4                     7                   4
ExampleMatField(7)              4                    18                   4
ExampleMatField(8)              4                    13                   4
ExampleMatField(9)              4                     7                   4
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3 An example application
 3.1 Number fields defined by matrices
 3.2 Number fields defined by a polynomial

3 An example application

In this section we outline two example computations with the functions of the previous chapter. The first example uses number fields defined by matrices and the second example considers number fields defined by a polynomial.

3.1 Number fields defined by matrices

gap> m1 := [ [ 1, 0, 0, -7 ], 
>            [ 7, 1, 0, -7 ], 
>            [ 0, 7, 1, -7 ], 
>            [ 0, 0, 7, -6 ] ];;

#
gap> m2 := [ [ 0, 0, -13, 14 ], 
>            [ -1, 0, -13, 1 ], 
>            [ 13, -1, -13, 1 ], 
>            [ 0, 13, -14, 1 ] ];;

#
gap> F := FieldByMatricesNC( [m1, m2] );
<rational matrix field of unknown degree>

#
gap> DegreeOverPrimeField(F);
4
gap> PrimitiveElement(F);
[ [ 0, -1, 1, 0 ], [ 0, -1, 0, 1 ], [ 0, -1, 0, 0 ], [ 1, -1, 0, 0 ] ]

#
gap> Basis(F);
Basis( <rational matrix field of degree 4>, 
[ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], 
  [ [ 0, 1, 0, 0 ], [ -1, 1, 1, 0 ], [ -1, 0, 1, 1 ], [ -1, 0, 0, 1 ] ], 
  [ [ 0, 0, 1, 0 ], [ -1, 0, 1, 1 ], [ -1, -1, 1, 1 ], [ 0, -1, 0, 1 ] ], 
  [ [ 0, 0, 0, 1 ], [ -1, 0, 0, 1 ], [ 0, -1, 0, 1 ], [ 0, 0, -1, 1 ] ] ] )

#
gap> MaximalOrderBasis(F);
Basis( <rational matrix field of degree 4>, 
[ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], 
  [ [ 0, -1, 1, 0 ], [ 0, -1, 0, 1 ], [ 0, -1, 0, 0 ], [ 1, -1, 0, 0 ] ], 
  [ [ 0, 0, 0, -1 ], [ 1, 0, 0, -1 ], [ 0, 1, 0, -1 ], [ 0, 0, 1, -1 ] ], 
  [ [ -1, 1, 0, 0 ], [ -1, 0, 1, 0 ], [ -1, 0, 0, 1 ], [ -1, 0, 0, 0 ] ] ] )

#
gap> U := UnitGroup(F);
<matrix group with 2 generators>

#
gap> u := GeneratorsOfGroup( U );;

#
gap> nat := IsomorphismPcpGroup(U);;
gap> H := Image(nat);
Pcp-group with orders [ 10, 0 ]
gap> ImageElm( nat, u[1] );
g1
gap> ImageElm( nat, u[2] );
g2
gap> ImageElm( nat, u[1]*u[2] );
g1*g2
gap> u[1] = PreImagesRepresentative(nat, GeneratorsOfGroup(H)[1] );
true

3.2 Number fields defined by a polynomial

gap> g := UnivariatePolynomial( Rationals, [ 16, 64, -28, -4, 1 ] );
x^4-4*x^3-28*x^2+64*x+16

#
gap> F := FieldByPolynomialNC(g);
<algebraic extension over the Rationals of degree 4>
gap> PrimitiveElement(F);
a
gap> MaximalOrderBasis(F);
Basis( <algebraic extension over the Rationals of degree 4>, 
[ !1, 1/2*a, 1/4*a^2, 1/56*a^3+1/14*a^2+1/14*a-2/7 ] )

#
gap> U := UnitGroup(F);
<group with 4 generators>

#
gap> natU := IsomorphismPcpGroup(U);;
gap> elms := List( [1..10], x-> Random(F) );;

#
gap>  PcpPresentationOfMultiplicativeSubgroup( F, elms );
Pcp-group with orders [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]

#
gap> isom := IsomorphismPcpGroup( F, elms );;
gap> y := RandomGroupElement( elms );;
gap> z := ImageElm( isom, y );;
gap> y = PreImagesRepresentative( isom, z );
true

#
gap> FactorsPolynomialAlgExt( F, g );
[ x_1+(-a), x_1+(a-2), x_1+(-1/7*a^3+3/7*a^2+31/7*a-40/7), 
  x_1+(1/7*a^3-3/7*a^2-31/7*a+26/7) ]
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alnuth-4.0.0/doc/chap3.txt0000644000000000000000000001315515201210200012224 0ustar00 3 An example application In this section we outline two example computations with the functions of the previous chapter. The first example uses number fields defined by matrices and the second example considers number fields defined by a polynomial. 3.1 Number fields defined by matrices  Example  gap> m1 := [ [ 1, 0, 0, -7 ],  >  [ 7, 1, 0, -7 ],  >  [ 0, 7, 1, -7 ],  >  [ 0, 0, 7, -6 ] ];;  # gap> m2 := [ [ 0, 0, -13, 14 ],  >  [ -1, 0, -13, 1 ],  >  [ 13, -1, -13, 1 ],  >  [ 0, 13, -14, 1 ] ];;  # gap> F := FieldByMatricesNC( [m1, m2] );   # gap> DegreeOverPrimeField(F); 4 gap> PrimitiveElement(F); [ [ 0, -1, 1, 0 ], [ 0, -1, 0, 1 ], [ 0, -1, 0, 0 ], [ 1, -1, 0, 0 ] ]  # gap> Basis(F); Basis( ,  [ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ],   [ [ 0, 1, 0, 0 ], [ -1, 1, 1, 0 ], [ -1, 0, 1, 1 ], [ -1, 0, 0, 1 ] ],   [ [ 0, 0, 1, 0 ], [ -1, 0, 1, 1 ], [ -1, -1, 1, 1 ], [ 0, -1, 0, 1 ] ],   [ [ 0, 0, 0, 1 ], [ -1, 0, 0, 1 ], [ 0, -1, 0, 1 ], [ 0, 0, -1, 1 ] ] ] )  # gap> MaximalOrderBasis(F); Basis( ,  [ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ],   [ [ 0, -1, 1, 0 ], [ 0, -1, 0, 1 ], [ 0, -1, 0, 0 ], [ 1, -1, 0, 0 ] ],   [ [ 0, 0, 0, -1 ], [ 1, 0, 0, -1 ], [ 0, 1, 0, -1 ], [ 0, 0, 1, -1 ] ],   [ [ -1, 1, 0, 0 ], [ -1, 0, 1, 0 ], [ -1, 0, 0, 1 ], [ -1, 0, 0, 0 ] ] ] )  # gap> U := UnitGroup(F);   # gap> u := GeneratorsOfGroup( U );;  # gap> nat := IsomorphismPcpGroup(U);; gap> H := Image(nat); Pcp-group with orders [ 10, 0 ] gap> ImageElm( nat, u[1] ); g1 gap> ImageElm( nat, u[2] ); g2 gap> ImageElm( nat, u[1]*u[2] ); g1*g2 gap> u[1] = PreImagesRepresentative(nat, GeneratorsOfGroup(H)[1] ); true  3.2 Number fields defined by a polynomial  Example  gap> g := UnivariatePolynomial( Rationals, [ 16, 64, -28, -4, 1 ] ); x^4-4*x^3-28*x^2+64*x+16  # gap> F := FieldByPolynomialNC(g);  gap> PrimitiveElement(F); a gap> MaximalOrderBasis(F); Basis( ,  [ !1, 1/2*a, 1/4*a^2, 1/56*a^3+1/14*a^2+1/14*a-2/7 ] )  # gap> U := UnitGroup(F);   # gap> natU := IsomorphismPcpGroup(U);; gap> elms := List( [1..10], x-> Random(F) );;  # gap>  PcpPresentationOfMultiplicativeSubgroup( F, elms ); Pcp-group with orders [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]  # gap> isom := IsomorphismPcpGroup( F, elms );; gap> y := RandomGroupElement( elms );; gap> z := ImageElm( isom, y );; gap> y = PreImagesRepresentative( isom, z ); true  # gap> FactorsPolynomialAlgExt( F, g ); [ x_1+(-a), x_1+(a-2), x_1+(-1/7*a^3+3/7*a^2+31/7*a-40/7),   x_1+(1/7*a^3-3/7*a^2-31/7*a+26/7) ]  alnuth-4.0.0/doc/chap3_mj.html0000644000000000000000000001765615201210200013051 0ustar00 GAP (Alnuth) - Chapter 3: An example application
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3 An example application
 3.1 Number fields defined by matrices
 3.2 Number fields defined by a polynomial

3 An example application

In this section we outline two example computations with the functions of the previous chapter. The first example uses number fields defined by matrices and the second example considers number fields defined by a polynomial.

3.1 Number fields defined by matrices

gap> m1 := [ [ 1, 0, 0, -7 ], 
>            [ 7, 1, 0, -7 ], 
>            [ 0, 7, 1, -7 ], 
>            [ 0, 0, 7, -6 ] ];;

#
gap> m2 := [ [ 0, 0, -13, 14 ], 
>            [ -1, 0, -13, 1 ], 
>            [ 13, -1, -13, 1 ], 
>            [ 0, 13, -14, 1 ] ];;

#
gap> F := FieldByMatricesNC( [m1, m2] );
<rational matrix field of unknown degree>

#
gap> DegreeOverPrimeField(F);
4
gap> PrimitiveElement(F);
[ [ 0, -1, 1, 0 ], [ 0, -1, 0, 1 ], [ 0, -1, 0, 0 ], [ 1, -1, 0, 0 ] ]

#
gap> Basis(F);
Basis( <rational matrix field of degree 4>, 
[ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], 
  [ [ 0, 1, 0, 0 ], [ -1, 1, 1, 0 ], [ -1, 0, 1, 1 ], [ -1, 0, 0, 1 ] ], 
  [ [ 0, 0, 1, 0 ], [ -1, 0, 1, 1 ], [ -1, -1, 1, 1 ], [ 0, -1, 0, 1 ] ], 
  [ [ 0, 0, 0, 1 ], [ -1, 0, 0, 1 ], [ 0, -1, 0, 1 ], [ 0, 0, -1, 1 ] ] ] )

#
gap> MaximalOrderBasis(F);
Basis( <rational matrix field of degree 4>, 
[ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], 
  [ [ 0, -1, 1, 0 ], [ 0, -1, 0, 1 ], [ 0, -1, 0, 0 ], [ 1, -1, 0, 0 ] ], 
  [ [ 0, 0, 0, -1 ], [ 1, 0, 0, -1 ], [ 0, 1, 0, -1 ], [ 0, 0, 1, -1 ] ], 
  [ [ -1, 1, 0, 0 ], [ -1, 0, 1, 0 ], [ -1, 0, 0, 1 ], [ -1, 0, 0, 0 ] ] ] )

#
gap> U := UnitGroup(F);
<matrix group with 2 generators>

#
gap> u := GeneratorsOfGroup( U );;

#
gap> nat := IsomorphismPcpGroup(U);;
gap> H := Image(nat);
Pcp-group with orders [ 10, 0 ]
gap> ImageElm( nat, u[1] );
g1
gap> ImageElm( nat, u[2] );
g2
gap> ImageElm( nat, u[1]*u[2] );
g1*g2
gap> u[1] = PreImagesRepresentative(nat, GeneratorsOfGroup(H)[1] );
true

3.2 Number fields defined by a polynomial

gap> g := UnivariatePolynomial( Rationals, [ 16, 64, -28, -4, 1 ] );
x^4-4*x^3-28*x^2+64*x+16

#
gap> F := FieldByPolynomialNC(g);
<algebraic extension over the Rationals of degree 4>
gap> PrimitiveElement(F);
a
gap> MaximalOrderBasis(F);
Basis( <algebraic extension over the Rationals of degree 4>, 
[ !1, 1/2*a, 1/4*a^2, 1/56*a^3+1/14*a^2+1/14*a-2/7 ] )

#
gap> U := UnitGroup(F);
<group with 4 generators>

#
gap> natU := IsomorphismPcpGroup(U);;
gap> elms := List( [1..10], x-> Random(F) );;

#
gap>  PcpPresentationOfMultiplicativeSubgroup( F, elms );
Pcp-group with orders [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]

#
gap> isom := IsomorphismPcpGroup( F, elms );;
gap> y := RandomGroupElement( elms );;
gap> z := ImageElm( isom, y );;
gap> y = PreImagesRepresentative( isom, z );
true

#
gap> FactorsPolynomialAlgExt( F, g );
[ x_1+(-a), x_1+(a-2), x_1+(-1/7*a^3+3/7*a^2+31/7*a-40/7), 
  x_1+(1/7*a^3-3/7*a^2-31/7*a+26/7) ]
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4 Installation
 4.1 Installing Alnuth
 4.2 Getting PARI/GP
 4.3 Adjust the path of the executable for GP

  4.3-1 PariVersion
  4.3-2 SetAlnuthExternalExecutable
  4.3-3 SetAlnuthExternalExecutablePermanently
  4.3-4 RestoreAlnuthExternalExecutablePermanently
 4.4 Loading and testing the package

4 Installation

This package provides an interface between GAP and either PARI/GP or OSCAR. When Alnuth is loaded in GAP running inside OSCAR, it automatically uses OSCAR. Otherwise, Alnuth uses PARI/GP, which has to be obtained and installed independently of this package. Alnuth works with PARI/GP Version 2.5 or higher.

4.1 Installing Alnuth

The package Alnuth is part of the standard distribution of GAP so that in most cases there is no need to install it separately. If you are using GAP inside OSCAR, then Alnuth is already bundled and automatically uses OSCAR, so no further installation is required. Otherwise, to use Alnuth you need to have PARI/GP installed. See the following section for information on PARI/GP.

In case you want to update Alnuth independently of your main GAP installation or if you are interested in an old version of Alnuth interfacing to KANT/KASH you can find all released versions of Alnuth in the form of gzipped tar-archives at https://github.com/gap-packages/alnuth/releases

There are two ways of installing a GAP package. If you have permission to add files to the installation of GAP on your system you may install Alnuth into the pkg subdirectory of the GAP installation tree. Otherwise you may install Alnuth in a private pkg directory (for details see Subsections Reference: Installing a GAP Package and Reference: GAP Root Directories in the GAP reference manual).

To install the latest version of Alnuth download one of the archives alnuth.tar.gz, move it to the directory pkg in which you want to install, and unpack the archive. If you are using the command line you can unpack with the command tar xzf alnuth.tar.gz.

4.2 Getting PARI/GP

Using Alnuth outside OSCAR requires an installation of PARI/GP in Version 2.5 or higher. The software PARI/GP is freely available at https://pari.math.u-bordeaux.fr/

Note that the place where PARI/GP is located in your system is independent of the place where Alnuth is installed.

  1. Installing under Linux

    In many Linux distributions PARI/GP can be installed via the software package manager, but this might sometimes be an older version which cannot be used together with Alnuth. (Starting GP from the command line with the option --version-short will show you the version number.)

    If you install PARI/GP from source make sure you install with GMP support for better performance and complete the installation with make install so that you can start GP by just calling gp from the command line.

  2. Installing under Windows

    For Windows it is sufficient to get the basic GP binary which can be found at https://pari.math.u-bordeaux.fr/download.html

4.3 Adjust the path of the executable for GP

When Alnuth is used outside OSCAR, it needs to know where the executable for GP is. In the default setting Alnuth looks for an executable program named gp in the search paths of the system. More precisely, for a file gp inside one of the directories in the list returned by DirectoriesSystemPrograms() (called in a GAP session).

Under Linux the default setting should work with a standard installation of PARI/GP.

For the default setting to work under Windows the downloaded executable file, for example gp-2-5-0.exe has to be renamed to gp.exe and moved to one of the directories listed by DirectoriesSystemPrograms() (Ignore the leading cygdrive in each path name and note that the single letter specifies the drive, for example /cygdrive/c/Windows/ denotes the folder Windows on drive C:).

To check whether an executable of GP in Version 2.5 or higher is available with the default setting, you can use the function

4.3-1 PariVersion
‣ PariVersion( )( function )

which prints the version number, if the user preference PariGpPath refers to a usable GP executable.

If you cannot use the default setting for you purpose, you have several ways to configure Alnuth.

The recommended approach is to use the GAP user preference system:

gap> SetUserPreference("alnuth", "PariGpPath", "/home/my/pari-2.5.0/gp");
gap> SetUserPreference("alnuth", "PariStackSize", 256);
gap> WriteGapIniFile();

This changes the current session immediately. WriteGapIniFile() writes the current non-default settings to your personal gap.ini, so they are used again in future GAP sessions. See Reference: Configuring User preferences for background on the user preference system.

The user preferences relevant for Alnuth are shown by ShowUserPreferences("alnuth");. The most important ones are PariGpPath, PariGpOptions, PariStackSize, and PrimitiveElementTrials.

In older versions of Alnuth configuration was done via the global variables AL_EXECUTABLE, AL_PATH, AL_OPTIONS, AL_STACKSIZE, and PRIM_TEST. These globals are deprecated, and they trigger a warning when used. The globals except for AL_PATH still serve as a fallback if the corresponding user preference is unusable. Their replacements are as follows:

AL_EXECUTABLE

use the user preference PariGpPath

AL_OPTIONS

use the user preference PariGpOptions

AL_STACKSIZE

use the user preference PariStackSize

PRIM_TEST

use the user preference PrimitiveElementTrials

AL_PATH

has been removed; Alnuth now always uses the bundled GP helper files.

For backwards compatibility, the following functions still exist after loading the package (see Section 4.4), but they are deprecated wrappers around the user preference system.

4.3-2 SetAlnuthExternalExecutable
‣ SetAlnuthExternalExecutable( path )( function )

sets the user preference PariGpPath for the current GAP session. Depending on your installation of PARI/GP and your operating system the string path can be either the command to start GP in a terminal (for example gp) or the complete path to the executable of GP. The function is deprecated; new code should call SetUserPreference("alnuth", "PariGpPath", path); directly.

The function returns the stored preference value.

4.3-3 SetAlnuthExternalExecutablePermanently
‣ SetAlnuthExternalExecutablePermanently( path )( function )

does the same as SetAlnuthExternalExecutable and then calls WriteGapIniFile() to persist the current user preference settings in your personal gap.ini. The function is deprecated; new code should call SetUserPreference("alnuth", "PariGpPath", path); WriteGapIniFile(); instead.

4.3-4 RestoreAlnuthExternalExecutablePermanently
‣ RestoreAlnuthExternalExecutablePermanently( )( function )

restores the default value of PariGpPath and then calls WriteGapIniFile(). The function is deprecated; new code should set PariGpPath to the desired default command or path and then call WriteGapIniFile() directly.

4.4 Loading and testing the package

If Alnuth is not loaded when GAP is started you have to request it explicitly to use it. This is done by calling LoadPackage("alnuth"); in a GAP session. If Alnuth had not been loaded already a short banner will be displayed.

gap> LoadPackage("alnuth");
Loading Alnuth 4.0.0 (ALgebraic NUmber THeory and an interface to PARI/GP and OSCAR)
by Björn Assmann,
   Andreas Distler (a.distler@tu-bs.de), and
   Bettina Eick (http://www.iaa.tu-bs.de/beick).
maintained by:
   Max Horn (https://www.quendi.de/math) and
   The GAP Team (support@gap-system.org).
Homepage: https://gap-packages.github.io/alnuth
Report issues at https://github.com/gap-packages/alnuth/issues
true
gap>

To load a certain version of Alnuth you can specify the version number as second argument in the call to LoadPackage. (See LoadPackage (Reference: LoadPackage) in the GAP reference manual or type ?LoadPackage within a GAP session).

Once the package is loaded, it is possible to check the correct installation running a short test by calling ReadPackage("Alnuth", "tst/testinstall.tst");.

gap> ReadPackage("Alnuth", "tst/testinstall.g");
Architecture: aarch64-apple-darwin21.4.0-default64-kv8

testing: GAPROOT/pkg/alnuth/tst/ALNUTH.tst
      66 ms (33 ms GC) and 11.0MB allocated for GAPROOT/pkg/alnuth/tst/ALNUTH.tst
testing: GAPROOT/pkg/alnuth/tst/version.tst
      21 ms (21 ms GC) and 29.6KB allocated for GAPROOT/pkg/alnuth/tst/version.tst
-----------------------------------
total        87 ms (54 ms GC) and 11.0MB allocated
              0 failures in 2 files

#I  No errors detected while testing
gap>

The architecture, timings and memory usage will usually differ; other discrepancies in the output indicate some problem.

If the test suite runs into an error in the first part, which when running outside OSCAR verifies the availability of PARI/GP, check your installation of PARI/GP and consult the last chapter of the documentation of Alnuth for more information.

If you find any bugs or have any suggestions or comments, we would very much appreciate it if you would let us know by submitting an issue at the Alnuth issue tracker on GitHub https://github.com/gap-packages/alnuth/issues or by writing an mail to support@gap-system.org.

 

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alnuth-4.0.0/doc/chap4.txt0000644000000000000000000003110015201210200012213 0ustar00 4 Installation This package provides an interface between GAP and either PARI/GP or OSCAR. When Alnuth is loaded in GAP running inside OSCAR, it automatically uses OSCAR. Otherwise, Alnuth uses PARI/GP, which has to be obtained and installed independently of this package. Alnuth works with PARI/GP Version 2.5 or higher. 4.1 Installing Alnuth The package Alnuth is part of the standard distribution of GAP so that in most cases there is no need to install it separately. If you are using GAP inside OSCAR, then Alnuth is already bundled and automatically uses OSCAR, so no further installation is required. Otherwise, to use Alnuth you need to have PARI/GP installed. See the following section for information on PARI/GP. In case you want to update Alnuth independently of your main GAP installation or if you are interested in an old version of Alnuth interfacing to KANT/KASH you can find all released versions of Alnuth in the form of gzipped tar-archives at https://github.com/gap-packages/alnuth/releases There are two ways of installing a GAP package. If you have permission to add files to the installation of GAP on your system you may install Alnuth into the pkg subdirectory of the GAP installation tree. Otherwise you may install Alnuth in a private pkg directory (for details see Subsections 'Reference: Installing a GAP Package' and 'Reference: GAP Root Directories' in the GAP reference manual). To install the latest version of Alnuth download one of the archives alnuth.tar.gz, move it to the directory pkg in which you want to install, and unpack the archive. If you are using the command line you can unpack with the command tar xzf alnuth.tar.gz. 4.2 Getting PARI/GP Using Alnuth outside OSCAR requires an installation of PARI/GP in Version 2.5 or higher. The software PARI/GP is freely available at https://pari.math.u-bordeaux.fr/ Note that the place where PARI/GP is located in your system is independent of the place where Alnuth is installed. 1 Installing under Linux In many Linux distributions PARI/GP can be installed via the software package manager, but this might sometimes be an older version which cannot be used together with Alnuth. (Starting GP from the command line with the option --version-short will show you the version number.) If you install PARI/GP from source make sure you install with GMP support for better performance and complete the installation with make install so that you can start GP by just calling gp from the command line. 2 Installing under Windows For Windows it is sufficient to get the basic GP binary which can be found at https://pari.math.u-bordeaux.fr/download.html 4.3 Adjust the path of the executable for GP When Alnuth is used outside OSCAR, it needs to know where the executable for GP is. In the default setting Alnuth looks for an executable program named gp in the search paths of the system. More precisely, for a file gp inside one of the directories in the list returned by DirectoriesSystemPrograms() (called in a GAP session). Under Linux the default setting should work with a standard installation of PARI/GP. For the default setting to work under Windows the downloaded executable file, for example gp-2-5-0.exe has to be renamed to gp.exe and moved to one of the directories listed by DirectoriesSystemPrograms() (Ignore the leading cygdrive in each path name and note that the single letter specifies the drive, for example /cygdrive/c/Windows/ denotes the folder Windows on drive C:). To check whether an executable of GP in Version 2.5 or higher is available with the default setting, you can use the function 4.3-1 PariVersion PariVersion( )  function which prints the version number, if the user preference PariGpPath refers to a usable GP executable. If you cannot use the default setting for you purpose, you have several ways to configure Alnuth. The recommended approach is to use the GAP user preference system:  Example  gap> SetUserPreference("alnuth", "PariGpPath", "/home/my/pari-2.5.0/gp"); gap> SetUserPreference("alnuth", "PariStackSize", 256); gap> WriteGapIniFile();  This changes the current session immediately. WriteGapIniFile() writes the current non-default settings to your personal gap.ini, so they are used again in future GAP sessions. See 'Reference: Configuring User preferences' for background on the user preference system. The user preferences relevant for Alnuth are shown by ShowUserPreferences("alnuth");. The most important ones are PariGpPath, PariGpOptions, PariStackSize, and PrimitiveElementTrials. In older versions of Alnuth configuration was done via the global variables AL_EXECUTABLE, AL_PATH, AL_OPTIONS, AL_STACKSIZE, and PRIM_TEST. These globals are deprecated, and they trigger a warning when used. The globals except for AL_PATH still serve as a fallback if the corresponding user preference is unusable. Their replacements are as follows: AL_EXECUTABLE use the user preference PariGpPath AL_OPTIONS use the user preference PariGpOptions AL_STACKSIZE use the user preference PariStackSize PRIM_TEST use the user preference PrimitiveElementTrials AL_PATH has been removed; Alnuth now always uses the bundled GP helper files. For backwards compatibility, the following functions still exist after loading the package (see Section 4.4), but they are deprecated wrappers around the user preference system. 4.3-2 SetAlnuthExternalExecutable SetAlnuthExternalExecutable( path )  function sets the user preference PariGpPath for the current GAP session. Depending on your installation of PARI/GP and your operating system the string path can be either the command to start GP in a terminal (for example gp) or the complete path to the executable of GP. The function is deprecated; new code should call SetUserPreference("alnuth", "PariGpPath", path); directly. The function returns the stored preference value. 4.3-3 SetAlnuthExternalExecutablePermanently SetAlnuthExternalExecutablePermanently( path )  function does the same as SetAlnuthExternalExecutable and then calls WriteGapIniFile() to persist the current user preference settings in your personal gap.ini. The function is deprecated; new code should call SetUserPreference("alnuth", "PariGpPath", path); WriteGapIniFile(); instead. 4.3-4 RestoreAlnuthExternalExecutablePermanently RestoreAlnuthExternalExecutablePermanently( )  function restores the default value of PariGpPath and then calls WriteGapIniFile(). The function is deprecated; new code should set PariGpPath to the desired default command or path and then call WriteGapIniFile() directly. 4.4 Loading and testing the package If Alnuth is not loaded when GAP is started you have to request it explicitly to use it. This is done by calling LoadPackage("alnuth"); in a GAP session. If Alnuth had not been loaded already a short banner will be displayed.  Example  gap> LoadPackage("alnuth"); Loading Alnuth 4.0.0 (ALgebraic NUmber THeory and an interface to PARI/GP and OSCAR) by Björn Assmann,  Andreas Distler (a.distler@tu-bs.de), and  Bettina Eick (http://www.iaa.tu-bs.de/beick). maintained by:  Max Horn (https://www.quendi.de/math) and  The GAP Team (support@gap-system.org). Homepage: https://gap-packages.github.io/alnuth Report issues at https://github.com/gap-packages/alnuth/issues true gap>  To load a certain version of Alnuth you can specify the version number as second argument in the call to LoadPackage. (See LoadPackage (Reference: LoadPackage) in the GAP reference manual or type ?LoadPackage within a GAP session). Once the package is loaded, it is possible to check the correct installation running a short test by calling ReadPackage("Alnuth", "tst/testinstall.tst");.  Example  gap> ReadPackage("Alnuth", "tst/testinstall.g"); Architecture: aarch64-apple-darwin21.4.0-default64-kv8  testing: GAPROOT/pkg/alnuth/tst/ALNUTH.tst  66 ms (33 ms GC) and 11.0MB allocated for GAPROOT/pkg/alnuth/tst/ALNUTH.tst testing: GAPROOT/pkg/alnuth/tst/version.tst  21 ms (21 ms GC) and 29.6KB allocated for GAPROOT/pkg/alnuth/tst/version.tst ----------------------------------- total 87 ms (54 ms GC) and 11.0MB allocated  0 failures in 2 files  #I No errors detected while testing gap>  The architecture, timings and memory usage will usually differ; other discrepancies in the output indicate some problem. If the test suite runs into an error in the first part, which when running outside OSCAR verifies the availability of PARI/GP, check your installation of PARI/GP and consult the last chapter of the documentation of Alnuth for more information. If you find any bugs or have any suggestions or comments, we would very much appreciate it if you would let us know by submitting an issue at the Alnuth issue tracker on GitHub https://github.com/gap-packages/alnuth/issues or by writing an mail to mailto:support@gap-system.org.   alnuth-4.0.0/doc/chap4_mj.html0000644000000000000000000004405615201210200013044 0ustar00 GAP (Alnuth) - Chapter 4: Installation
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4 Installation
 4.1 Installing Alnuth
 4.2 Getting PARI/GP
 4.3 Adjust the path of the executable for GP

  4.3-1 PariVersion
  4.3-2 SetAlnuthExternalExecutable
  4.3-3 SetAlnuthExternalExecutablePermanently
  4.3-4 RestoreAlnuthExternalExecutablePermanently
 4.4 Loading and testing the package

4 Installation

This package provides an interface between GAP and either PARI/GP or OSCAR. When Alnuth is loaded in GAP running inside OSCAR, it automatically uses OSCAR. Otherwise, Alnuth uses PARI/GP, which has to be obtained and installed independently of this package. Alnuth works with PARI/GP Version 2.5 or higher.

4.1 Installing Alnuth

The package Alnuth is part of the standard distribution of GAP so that in most cases there is no need to install it separately. If you are using GAP inside OSCAR, then Alnuth is already bundled and automatically uses OSCAR, so no further installation is required. Otherwise, to use Alnuth you need to have PARI/GP installed. See the following section for information on PARI/GP.

In case you want to update Alnuth independently of your main GAP installation or if you are interested in an old version of Alnuth interfacing to KANT/KASH you can find all released versions of Alnuth in the form of gzipped tar-archives at https://github.com/gap-packages/alnuth/releases

There are two ways of installing a GAP package. If you have permission to add files to the installation of GAP on your system you may install Alnuth into the pkg subdirectory of the GAP installation tree. Otherwise you may install Alnuth in a private pkg directory (for details see Subsections Reference: Installing a GAP Package and Reference: GAP Root Directories in the GAP reference manual).

To install the latest version of Alnuth download one of the archives alnuth.tar.gz, move it to the directory pkg in which you want to install, and unpack the archive. If you are using the command line you can unpack with the command tar xzf alnuth.tar.gz.

4.2 Getting PARI/GP

Using Alnuth outside OSCAR requires an installation of PARI/GP in Version 2.5 or higher. The software PARI/GP is freely available at https://pari.math.u-bordeaux.fr/

Note that the place where PARI/GP is located in your system is independent of the place where Alnuth is installed.

  1. Installing under Linux

    In many Linux distributions PARI/GP can be installed via the software package manager, but this might sometimes be an older version which cannot be used together with Alnuth. (Starting GP from the command line with the option --version-short will show you the version number.)

    If you install PARI/GP from source make sure you install with GMP support for better performance and complete the installation with make install so that you can start GP by just calling gp from the command line.

  2. Installing under Windows

    For Windows it is sufficient to get the basic GP binary which can be found at https://pari.math.u-bordeaux.fr/download.html

4.3 Adjust the path of the executable for GP

When Alnuth is used outside OSCAR, it needs to know where the executable for GP is. In the default setting Alnuth looks for an executable program named gp in the search paths of the system. More precisely, for a file gp inside one of the directories in the list returned by DirectoriesSystemPrograms() (called in a GAP session).

Under Linux the default setting should work with a standard installation of PARI/GP.

For the default setting to work under Windows the downloaded executable file, for example gp-2-5-0.exe has to be renamed to gp.exe and moved to one of the directories listed by DirectoriesSystemPrograms() (Ignore the leading cygdrive in each path name and note that the single letter specifies the drive, for example /cygdrive/c/Windows/ denotes the folder Windows on drive C:).

To check whether an executable of GP in Version 2.5 or higher is available with the default setting, you can use the function

4.3-1 PariVersion
‣ PariVersion( )( function )

which prints the version number, if the user preference PariGpPath refers to a usable GP executable.

If you cannot use the default setting for you purpose, you have several ways to configure Alnuth.

The recommended approach is to use the GAP user preference system:

gap> SetUserPreference("alnuth", "PariGpPath", "/home/my/pari-2.5.0/gp");
gap> SetUserPreference("alnuth", "PariStackSize", 256);
gap> WriteGapIniFile();

This changes the current session immediately. WriteGapIniFile() writes the current non-default settings to your personal gap.ini, so they are used again in future GAP sessions. See Reference: Configuring User preferences for background on the user preference system.

The user preferences relevant for Alnuth are shown by ShowUserPreferences("alnuth");. The most important ones are PariGpPath, PariGpOptions, PariStackSize, and PrimitiveElementTrials.

In older versions of Alnuth configuration was done via the global variables AL_EXECUTABLE, AL_PATH, AL_OPTIONS, AL_STACKSIZE, and PRIM_TEST. These globals are deprecated, and they trigger a warning when used. The globals except for AL_PATH still serve as a fallback if the corresponding user preference is unusable. Their replacements are as follows:

AL_EXECUTABLE

use the user preference PariGpPath

AL_OPTIONS

use the user preference PariGpOptions

AL_STACKSIZE

use the user preference PariStackSize

PRIM_TEST

use the user preference PrimitiveElementTrials

AL_PATH

has been removed; Alnuth now always uses the bundled GP helper files.

For backwards compatibility, the following functions still exist after loading the package (see Section 4.4), but they are deprecated wrappers around the user preference system.

4.3-2 SetAlnuthExternalExecutable
‣ SetAlnuthExternalExecutable( path )( function )

sets the user preference PariGpPath for the current GAP session. Depending on your installation of PARI/GP and your operating system the string path can be either the command to start GP in a terminal (for example gp) or the complete path to the executable of GP. The function is deprecated; new code should call SetUserPreference("alnuth", "PariGpPath", path); directly.

The function returns the stored preference value.

4.3-3 SetAlnuthExternalExecutablePermanently
‣ SetAlnuthExternalExecutablePermanently( path )( function )

does the same as SetAlnuthExternalExecutable and then calls WriteGapIniFile() to persist the current user preference settings in your personal gap.ini. The function is deprecated; new code should call SetUserPreference("alnuth", "PariGpPath", path); WriteGapIniFile(); instead.

4.3-4 RestoreAlnuthExternalExecutablePermanently
‣ RestoreAlnuthExternalExecutablePermanently( )( function )

restores the default value of PariGpPath and then calls WriteGapIniFile(). The function is deprecated; new code should set PariGpPath to the desired default command or path and then call WriteGapIniFile() directly.

4.4 Loading and testing the package

If Alnuth is not loaded when GAP is started you have to request it explicitly to use it. This is done by calling LoadPackage("alnuth"); in a GAP session. If Alnuth had not been loaded already a short banner will be displayed.

gap> LoadPackage("alnuth");
Loading Alnuth 4.0.0 (ALgebraic NUmber THeory and an interface to PARI/GP and OSCAR)
by Björn Assmann,
   Andreas Distler (a.distler@tu-bs.de), and
   Bettina Eick (http://www.iaa.tu-bs.de/beick).
maintained by:
   Max Horn (https://www.quendi.de/math) and
   The GAP Team (support@gap-system.org).
Homepage: https://gap-packages.github.io/alnuth
Report issues at https://github.com/gap-packages/alnuth/issues
true
gap>

To load a certain version of Alnuth you can specify the version number as second argument in the call to LoadPackage. (See LoadPackage (Reference: LoadPackage) in the GAP reference manual or type ?LoadPackage within a GAP session).

Once the package is loaded, it is possible to check the correct installation running a short test by calling ReadPackage("Alnuth", "tst/testinstall.tst");.

gap> ReadPackage("Alnuth", "tst/testinstall.g");
Architecture: aarch64-apple-darwin21.4.0-default64-kv8

testing: GAPROOT/pkg/alnuth/tst/ALNUTH.tst
      66 ms (33 ms GC) and 11.0MB allocated for GAPROOT/pkg/alnuth/tst/ALNUTH.tst
testing: GAPROOT/pkg/alnuth/tst/version.tst
      21 ms (21 ms GC) and 29.6KB allocated for GAPROOT/pkg/alnuth/tst/version.tst
-----------------------------------
total        87 ms (54 ms GC) and 11.0MB allocated
              0 failures in 2 files

#I  No errors detected while testing
gap>

The architecture, timings and memory usage will usually differ; other discrepancies in the output indicate some problem.

If the test suite runs into an error in the first part, which when running outside OSCAR verifies the availability of PARI/GP, check your installation of PARI/GP and consult the last chapter of the documentation of Alnuth for more information.

If you find any bugs or have any suggestions or comments, we would very much appreciate it if you would let us know by submitting an issue at the Alnuth issue tracker on GitHub https://github.com/gap-packages/alnuth/issues or by writing an mail to support@gap-system.org.

 

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References

[Coh93] Cohen, H., A course in computational algebraic number theory, Springer-Verlag, New York, Heidelberg, Berlin (1993).

[DEF+25] (Decker, W., Eder, C., Fieker, C., Horn, M. and Joswig, M., Eds.), The Computer Algebra System OSCAR: Algorithms and Examples, Springer, 1 edition, Algorithms and Computation in Mathematics, 32 (2025), (10.1007/978-3-031-62127-7).

[EHN11] Eick, B., Horn, M. and Nickel, W., Polycyclic - a GAP package (2011).

[OSC24] OSCAR -- Open Source Computer Algebra Research system, Version 1.0.0, The OSCAR Team (2024).

[PAR11] PARI/GP, version 2.5.0, {The PARI~Group}, Bordeaux (2011).

[Poh93] Pohst, M. E., Computational Algebraic Number Theory, Birkhäuser, DMV Seminar, 21 (1993).

[PZ89] Pohst, M. and Zassenhaus, H., Algorithmic algebraic number theory, Cambridge University Press (1989).

[ST79] Stuart, I. N. and Tall, D. O., Algebraic number theory, Chapman and Hall (1979).

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alnuth-4.0.0/doc/chapBib.txt0000644000000000000000000000247115201210200012555 0ustar00 References [Coh93] Cohen, H., A course in computational algebraic number theory, Springer-Verlag, New York, Heidelberg, Berlin (1993). [DEF+25] (Decker, W., Eder, C., Fieker, C., Horn, M. and Joswig, M., Eds.), The Computer Algebra System OSCAR: Algorithms and Examples, Springer, 1 edition, Algorithms and Computation in Mathematics, 32 (2025) (https://doi.org/10.1007/978-3-031-62127-7). [EHN11] Eick, B., Horn, M. and Nickel, W., Polycyclic - a GAP package (2011). [OSC24] OSCAR -- Open Source Computer Algebra Research system, Version 1.0.0, The OSCAR Team (2024). [PAR11] PARI/GP, version 2.5.0, {The PARI~Group}, Bordeaux (2011). [Poh93] Pohst, M. E., Computational Algebraic Number Theory, Birkhäuser, DMV Seminar, 21 (1993). [PZ89] Pohst, M. and Zassenhaus, H., Algorithmic algebraic number theory, Cambridge University Press (1989). [ST79] Stuart, I. N. and Tall, D. O., Algebraic number theory, Chapman and Hall (1979).  alnuth-4.0.0/doc/chapBib_mj.html0000644000000000000000000001230115201210200013361 0ustar00 GAP (Alnuth) - References
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References

[Coh93] Cohen, H., A course in computational algebraic number theory, Springer-Verlag, New York, Heidelberg, Berlin (1993).

[DEF+25] (Decker, W., Eder, C., Fieker, C., Horn, M. and Joswig, M., Eds.), The Computer Algebra System OSCAR: Algorithms and Examples, Springer, 1 edition, Algorithms and Computation in Mathematics, 32 (2025), (10.1007/978-3-031-62127-7).

[EHN11] Eick, B., Horn, M. and Nickel, W., Polycyclic - a GAP package (2011).

[OSC24] OSCAR -- Open Source Computer Algebra Research system, Version 1.0.0, The OSCAR Team (2024).

[PAR11] PARI/GP, version 2.5.0, {The PARI~Group}, Bordeaux (2011).

[Poh93] Pohst, M. E., Computational Algebraic Number Theory, Birkhäuser, DMV Seminar, 21 (1993).

[PZ89] Pohst, M. and Zassenhaus, H., Algorithmic algebraic number theory, Cambridge University Press (1989).

[ST79] Stuart, I. N. and Tall, D. O., Algebraic number theory, Chapman and Hall (1979).

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Index

DefiningPolynomial 2.2-1
DegreeOverPrimeField 2.2-3
EquationOrderBasis 2.2-4
ExampleMatField 2.6-1
ExponentsOfUnits 2.2-7
FactorsPolynomialAlgExt 2.5-1
FactorsPolynomialAlnuth 2.5-2
FieldByMatrices 2.1-1
FieldByMatricesNC 2.1-1
FieldByMatrixBasis 2.1-2
FieldByMatrixBasisNC 2.1-2
FieldByPolynomial 2.1-3
FieldByPolynomialNC 2.1-3
IntersectionOfUnitSubgroups 2.4-2
IsCyclotomicField 2.2-8
IsIntegerOfNumberField 2.2-4
IsomorphismPcpGroup 2.3-1
IsPrimitiveElementOfNumberField 2.2-2
IsUnitOfNumberField 2.2-6
MaximalOrderBasis 2.2-4
NormCosetsOfNumberField 2.2-9
PariVersion 4.3-1
PcpPresentationOfMultiplicativeSubgroup 2.3-1
PrimitiveElement 2.2-1
RelationLattice 2.3-2
RelationLatticeOfUnits 2.4-1
RestoreAlnuthExternalExecutablePermanently 4.3-4
SetAlnuthExternalExecutable 4.3-2
SetAlnuthExternalExecutablePermanently 4.3-3
UnitGroup 2.2-5

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alnuth-4.0.0/doc/chapInd.txt0000644000000000000000000000245415201210200012574 0ustar00 Index DefiningPolynomial 2.2-1 DegreeOverPrimeField 2.2-3 EquationOrderBasis 2.2-4 ExampleMatField 2.6-1 ExponentsOfUnits 2.2-7 FactorsPolynomialAlgExt 2.5-1 FactorsPolynomialAlnuth 2.5-2 FieldByMatrices 2.1-1 FieldByMatricesNC 2.1-1 FieldByMatrixBasis 2.1-2 FieldByMatrixBasisNC 2.1-2 FieldByPolynomial 2.1-3 FieldByPolynomialNC 2.1-3 IntersectionOfUnitSubgroups 2.4-2 IsCyclotomicField 2.2-8 IsIntegerOfNumberField 2.2-4 IsomorphismPcpGroup 2.3-1 IsPrimitiveElementOfNumberField 2.2-2 IsUnitOfNumberField 2.2-6 MaximalOrderBasis 2.2-4 NormCosetsOfNumberField 2.2-9 PariVersion 4.3-1 PcpPresentationOfMultiplicativeSubgroup 2.3-1 PrimitiveElement 2.2-1 RelationLattice 2.3-2 RelationLatticeOfUnits 2.4-1 RestoreAlnuthExternalExecutablePermanently 4.3-4 SetAlnuthExternalExecutable 4.3-2 SetAlnuthExternalExecutablePermanently 4.3-3 UnitGroup 2.2-5 ------------------------------------------------------- alnuth-4.0.0/doc/chapInd_mj.html0000644000000000000000000001237215201210200013407 0ustar00 GAP (Alnuth) - Index
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Index

DefiningPolynomial 2.2-1
DegreeOverPrimeField 2.2-3
EquationOrderBasis 2.2-4
ExampleMatField 2.6-1
ExponentsOfUnits 2.2-7
FactorsPolynomialAlgExt 2.5-1
FactorsPolynomialAlnuth 2.5-2
FieldByMatrices 2.1-1
FieldByMatricesNC 2.1-1
FieldByMatrixBasis 2.1-2
FieldByMatrixBasisNC 2.1-2
FieldByPolynomial 2.1-3
FieldByPolynomialNC 2.1-3
IntersectionOfUnitSubgroups 2.4-2
IsCyclotomicField 2.2-8
IsIntegerOfNumberField 2.2-4
IsomorphismPcpGroup 2.3-1
IsPrimitiveElementOfNumberField 2.2-2
IsUnitOfNumberField 2.2-6
MaximalOrderBasis 2.2-4
NormCosetsOfNumberField 2.2-9
PariVersion 4.3-1
PcpPresentationOfMultiplicativeSubgroup 2.3-1
PrimitiveElement 2.2-1
RelationLattice 2.3-2
RelationLatticeOfUnits 2.4-1
RestoreAlnuthExternalExecutablePermanently 4.3-4
SetAlnuthExternalExecutable 4.3-2
SetAlnuthExternalExecutablePermanently 4.3-3
UnitGroup 2.2-5

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alnuth-4.0.0/doc/dark.css0000644000000000000000000000515115201210200012115 0ustar00/* dark.css Frank Luebeck */ /* Initial dark theme contributed by kiryph in issue #75. */ /* colors */ body { background: #121212; color: #eee; } a:link { color: #576cad; } a:visited { color: #576cad; } a:active { color: #eee; } a:hover { background: #eee; } pre { color: black; } tt, code { color: #eee; } /* layout for the definitions of functions, variables, ... */ div.func { background: #909090; } /* Example elements (for old converted manuals, now in div+pre */ table.example { background: #efefef; } /* becomes ... */ div.example { background: #efefef; color: black; } /* Links to chapters in all files at top and bottom. */ div.chlinktop { background: #22272e; border-color: #3a414a; color: #d7dde5; } div.chlinktop a:hover { background: #2f3742; } div.chlinkbot { background: #22272e; border-color: #3a414a; color: #d7dde5; } /* and this is for the "Top", "Prev", "Next" links */ div.chlinkprevnexttop { background: #22272e; border-color: #3a414a; color: #d7dde5; } div.chlinkprevnexttop a:hover { background: #2f3742; } div.chlinkprevnextbot { background: #22272e; border-color: #3a414a; color: #d7dde5; } div.chlinkprevnextbot a:hover { background: #2f3742; } div.ContChap div.ContSect:hover div.ContSSBlock { background: #eee; border-color: #666; color: #000; } div.ContSSBlock a:hover { background: #fff; } /* and here for the side menu of contents in the chapter files */ div.ChapSects a:hover { background: #eee; color: #000; } div.ChapSects div.ContSect:hover div.ContSSBlock { background: #b5b5b5; border-color: #666; color: #000; } div.ChapSects div.ContSect:hover div.ContSSBlock a:hover { background: #828282; } /* Table elements */ table.GAPDocTable { border-color: black; } table.GAPDocTable td, table.GAPDocTable th { border-color: #555; } table.GAPDocTablenoborder td, table.GAPDocTable th { border-color: #555; } /* Colors and fonts can be overwritten for some types of elements. */ /* Verb elements */ pre.normal { color: #eee; } /* Func-like elements and Ref to Func-like */ code.func { color: #eee; } /* K elements */ code.keyw { color: #983d3d; } /* F elements */ code.file { color: #8e4510; } /* Arg elements */ var.Arg { color: #060; } /* colors for ColorPrompt like examples */ span.GAPprompt { color: #000097; } span.GAPbrkprompt { color: #970000; } span.GAPinput { color: #970000; } /* Bib entries */ span.BibKey { color: #052; } /* for light and dark mode pictures */ .only-on-dark { display: block; } .only-on-light { display: none; } alnuth-4.0.0/doc/example.xml0000644000000000000000000000633015201210200012637 0ustar00 An example application In this section we outline two example computations with the functions of the previous chapter. The first example uses number fields defined by matrices and the second example considers number fields defined by a polynomial.
Number fields defined by matrices m1 := [ [ 1, 0, 0, -7 ], > [ 7, 1, 0, -7 ], > [ 0, 7, 1, -7 ], > [ 0, 0, 7, -6 ] ];; # gap> m2 := [ [ 0, 0, -13, 14 ], > [ -1, 0, -13, 1 ], > [ 13, -1, -13, 1 ], > [ 0, 13, -14, 1 ] ];; # gap> F := FieldByMatricesNC( [m1, m2] ); # gap> DegreeOverPrimeField(F); 4 gap> PrimitiveElement(F); [ [ 0, -1, 1, 0 ], [ 0, -1, 0, 1 ], [ 0, -1, 0, 0 ], [ 1, -1, 0, 0 ] ] # gap> Basis(F); Basis( , [ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 0, 1, 0, 0 ], [ -1, 1, 1, 0 ], [ -1, 0, 1, 1 ], [ -1, 0, 0, 1 ] ], [ [ 0, 0, 1, 0 ], [ -1, 0, 1, 1 ], [ -1, -1, 1, 1 ], [ 0, -1, 0, 1 ] ], [ [ 0, 0, 0, 1 ], [ -1, 0, 0, 1 ], [ 0, -1, 0, 1 ], [ 0, 0, -1, 1 ] ] ] ) # gap> MaximalOrderBasis(F); Basis( , [ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 0, -1, 1, 0 ], [ 0, -1, 0, 1 ], [ 0, -1, 0, 0 ], [ 1, -1, 0, 0 ] ], [ [ 0, 0, 0, -1 ], [ 1, 0, 0, -1 ], [ 0, 1, 0, -1 ], [ 0, 0, 1, -1 ] ], [ [ -1, 1, 0, 0 ], [ -1, 0, 1, 0 ], [ -1, 0, 0, 1 ], [ -1, 0, 0, 0 ] ] ] ) # gap> U := UnitGroup(F); # gap> u := GeneratorsOfGroup( U );; # gap> nat := IsomorphismPcpGroup(U);; gap> H := Image(nat); Pcp-group with orders [ 10, 0 ] gap> ImageElm( nat, u[1] ); g1 gap> ImageElm( nat, u[2] ); g2 gap> ImageElm( nat, u[1]*u[2] ); g1*g2 gap> u[1] = PreImagesRepresentative(nat, GeneratorsOfGroup(H)[1] ); true ]]>
Number fields defined by a polynomial g := UnivariatePolynomial( Rationals, [ 16, 64, -28, -4, 1 ] ); x^4-4*x^3-28*x^2+64*x+16 # gap> F := FieldByPolynomialNC(g); gap> PrimitiveElement(F); a gap> MaximalOrderBasis(F); Basis( , [ !1, 1/2*a, 1/4*a^2, 1/56*a^3+1/14*a^2+1/14*a-2/7 ] ) # gap> U := UnitGroup(F); # gap> natU := IsomorphismPcpGroup(U);; gap> elms := List( [1..10], x-> Random(F) );; # gap> PcpPresentationOfMultiplicativeSubgroup( F, elms ); Pcp-group with orders [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] # gap> isom := IsomorphismPcpGroup( F, elms );; gap> y := RandomGroupElement( elms );; gap> z := ImageElm( isom, y );; gap> y = PreImagesRepresentative( isom, z ); true # gap> FactorsPolynomialAlgExt( F, g ); [ x_1+(-a), x_1+(a-2), x_1+(-1/7*a^3+3/7*a^2+31/7*a-40/7), x_1+(1/7*a^3-3/7*a^2-31/7*a+26/7) ] ]]>
alnuth-4.0.0/doc/fields.xml0000644000000000000000000002430615201210200012455 0ustar00 Working with number fields An algebraic number field is a finite-dimensional extension of the rational numbers &QQ;. Such a number field has a primitive element and it can be defined by the minimal polynomial of this primitive element. Another important way to define an algebraic number field is by a set of rational matrices which generate a number field.
Creation of number fields We provide functions to create number fields defined by rational matrices or by rational polynomials. Creates a field generated by the rational matrices matrices. In the faster NC version, the function assumes that the input generates a field and there are no checks on this performed. Creates a field with basis matrices. The list matrices must consist of rational matrices which form a basis for a number field. In the faster NC version, the function assumes that the input is a matrix basis for a field and no checks are performed. Creates a field defined by polynomial. The polynomial polynomial must be an irreducible rational polynomial. In the faster NC version, no checks on the input are performed.
Methods for number fields We outline a number of functions for number fields. Computes a primitive element and a defining polynomial for the given number field. The defining polynomial is the minimal polynomial of the primitive element. Since F contains various primitive elements, PrimitiveElement tries to find a primitive element which has a minimal polynomial with small coefficients. Via the user preference PrimitiveElementTrials the user can decide how many primitive elements will be compared. The default value is 20. (See for more information about user preferences.) Checks if the given element generates the field. Returns the degree of F over the rationals. These functions return bases for the equation order or the maximal order of the number field F. Also, they allow to check if a given element is an integer in the given number field. determines the unit group of F.

Recall that the unit group of F is a finitely generated abelian group. The function IsomorphismPcpGroup from the &Polycyclic; package gives an isomorphism to a pcp group which can be used for various computations with the unit group. checks whether the element k is a unit in F. This function determines the exponent vectors of the elements in elms with respect to the generators of the unit group of F. If the unit group of F is not known, then the function computes this unit group also. Check whether F is cyclotomic. Returns a description for the set of all elements of norm norm in F. These elements can be written as a finite union of cosets of the unit group of F. The function returns coset representatives for these cosets.

Presentations of multiplicative subgroups Suppose that a finite number of invertible elements of a number field are given. Then these elements generate a finitely generated abelian group. However, it is a non-trivial task to provide a presentation for this abelian group. The most useful representation for such groups is as pcp group. Determine a pcp presentation for the multiplicative group of F\backslash\{0\} generated by elms and an isomorphism on this presentation. Note, that the method IsomorphismPcpGroup is defined in the &Polycyclic; package . We refer to the manual of this package for further background.

In the determination of the Pcp-presentation of a multiplicative subgroup generated by elms the relations between the elements in elms play an important role. Let elms=\{e_1,\dots,e_l\} be a finite subset of a field F. The relation lattice for elms is rl(elms):=\left\{(h_1,\dots,h_l) \in &ZZ;^l | e_1^{h_1} \cdots e_l^{h_l} = 1\right\} . Determines a generating set for the relation lattice of the field elements elms.

Methods to compute with subgroups of the unit group Determines a basis for the relation lattice of the units elms in triangularized form. Note that this method is more efficient than the method RelationLattice. The lists gen1 and gen2 are supposed to generate two subgroups U_1 and U_2 of the unit group of F. This function determines the intersection of U_1 with U_2. The result is returned as a list of vectors generating the lattice \{ e \in &ZZ;^n \mid g_1^{e_1} \cdots g_n^{e_n} \in U_2 \} for gen1 = [g_1, \ldots, g_n].

For efficiency reasons this function does not check the input and it may return wrong results if the input generators do not fulfil the requirements.

Factorisation of polynomials over a number field embeds the rational polynomial pol into the polynomial ring over the number field F, which has to be constructed by FieldByPolynomial or AlgebraicExtension, and returns the factorization of the embedded polynomial. By default a denotes the primitive element of the field one can obtain from PrimitiveElement(F), that is, a root of the defining polynomial of F. takes a polynomial pol defined over an algebraic extension of the Rationals and factors it using the external CAS (PARI/GP or OSCAR).

The alias FactorsPolynomialPari is provided for backwards compatibility. x := Indeterminate( Rationals, "x" );; gap> pol := 2*x^7+2*x^5+8*x^4+8*x^2; 2*x^7+2*x^5+8*x^4+8*x^2 gap> L := FieldByPolynomial( x^3-4 ); gap> y := Indeterminate( L, "y" );; gap> FactorsPolynomialAlgExt( L, pol ); [ !2*y, y, y+a, y^2+!1, y^2+(-a)*y+a^2 ] gap> FactorsPolynomialAlnuth( last[5] ); [ y^2+(-a)*y+a^2 ] ]]>

Examples This function returns some examples of fields generated by matrices. There are 9 such example fields provided and they can be obtained by assigning the input l to an integer between 1 and 9. Some of the properties of the examples are summarized in the following table.
alnuth-4.0.0/doc/install.xml0000644000000000000000000002455115201210200012657 0ustar00 Installation This package provides an interface between &GAP; and either PARI/GP or OSCAR. When &Alnuth; is loaded in &GAP; running inside OSCAR, it automatically uses OSCAR. Otherwise, &Alnuth; uses PARI/GP, which has to be obtained and installed independently of this package. &Alnuth; works with PARI/GP Version 2.5 or higher.
Installing Alnuth The package &Alnuth; is part of the standard distribution of &GAP; so that in most cases there is no need to install it separately. If you are using &GAP; inside OSCAR, then &Alnuth; is already bundled and automatically uses OSCAR, so no further installation is required. Otherwise, to use &Alnuth; you need to have PARI/GP installed. See the following section for information on PARI/GP.

In case you want to update &Alnuth; independently of your main &GAP; installation or if you are interested in an old version of &Alnuth; interfacing to KANT/KASH you can find all released versions of &Alnuth; in the form of gzipped tar-archives at https://github.com/gap-packages/alnuth/releases

There are two ways of installing a &GAP; package. If you have permission to add files to the installation of &GAP; on your system you may install &Alnuth; into the pkg subdirectory of the &GAP; installation tree. Otherwise you may install &Alnuth; in a private pkg directory (for details see Subsections and in the &GAP; reference manual).

To install the latest version of &Alnuth; download one of the archives alnuth.tar.gz, move it to the directory pkg in which you want to install, and unpack the archive. If you are using the command line you can unpack with the command tar xzf alnuth.tar.gz.

Getting PARI/GP Using &Alnuth; outside OSCAR requires an installation of PARI/GP in Version 2.5 or higher. The software PARI/GP is freely available at https://pari.math.u-bordeaux.fr/

Note that the place where PARI/GP is located in your system is independent of the place where &Alnuth; is installed.

Installing under Linux

In many Linux distributions PARI/GP can be installed via the software package manager, but this might sometimes be an older version which cannot be used together with &Alnuth;. (Starting GP from the command line with the option --version-short will show you the version number.)

If you install PARI/GP from source make sure you install with GMP support for better performance and complete the installation with make install so that you can start GP by just calling gp from the command line. Installing under Windows

For Windows it is sufficient to get the basic GP binary which can be found at https://pari.math.u-bordeaux.fr/download.html

Adjust the path of the executable for GP When &Alnuth; is used outside OSCAR, it needs to know where the executable for GP is. In the default setting &Alnuth; looks for an executable program named gp in the search paths of the system. More precisely, for a file gp inside one of the directories in the list returned by DirectoriesSystemPrograms() (called in a &GAP; session).

Under Linux the default setting should work with a standard installation of PARI/GP.

For the default setting to work under Windows the downloaded executable file, for example gp-2-5-0.exe has to be renamed to gp.exe and moved to one of the directories listed by DirectoriesSystemPrograms() (Ignore the leading cygdrive in each path name and note that the single letter specifies the drive, for example /cygdrive/c/Windows/ denotes the folder Windows on drive C:).

To check whether an executable of GP in Version 2.5 or higher is available with the default setting, you can use the function which prints the version number, if the user preference PariGpPath refers to a usable GP executable. If you cannot use the default setting for you purpose, you have several ways to configure Alnuth.

The recommended approach is to use the &GAP; user preference system: SetUserPreference("alnuth", "PariGpPath", "/home/my/pari-2.5.0/gp"); gap> SetUserPreference("alnuth", "PariStackSize", 256); gap> WriteGapIniFile(); ]]> This changes the current session immediately. WriteGapIniFile() writes the current non-default settings to your personal gap.ini, so they are used again in future &GAP; sessions. See for background on the user preference system.

The user preferences relevant for Alnuth are shown by ShowUserPreferences("alnuth");. The most important ones are PariGpPath, PariGpOptions, PariStackSize, and PrimitiveElementTrials.

In older versions of &Alnuth; configuration was done via the global variables AL_EXECUTABLE, AL_PATH, AL_OPTIONS, AL_STACKSIZE, and PRIM_TEST. These globals are deprecated, and they trigger a warning when used. The globals except for AL_PATH still serve as a fallback if the corresponding user preference is unusable. Their replacements are as follows: AL_EXECUTABLE use the user preference PariGpPath AL_OPTIONS use the user preference PariGpOptions AL_STACKSIZE use the user preference PariStackSize PRIM_TEST use the user preference PrimitiveElementTrials AL_PATH has been removed; Alnuth now always uses the bundled GP helper files.

For backwards compatibility, the following functions still exist after loading the package (see Section ), but they are deprecated wrappers around the user preference system. sets the user preference PariGpPath for the current &GAP; session. Depending on your installation of PARI/GP and your operating system the string path can be either the command to start GP in a terminal (for example gp) or the complete path to the executable of GP. The function is deprecated; new code should call SetUserPreference("alnuth", "PariGpPath", path); directly.

The function returns the stored preference value. does the same as SetAlnuthExternalExecutable and then calls WriteGapIniFile() to persist the current user preference settings in your personal gap.ini. The function is deprecated; new code should call SetUserPreference("alnuth", "PariGpPath", path); WriteGapIniFile(); instead. restores the default value of PariGpPath and then calls WriteGapIniFile(). The function is deprecated; new code should set PariGpPath to the desired default command or path and then call WriteGapIniFile() directly.

Loading and testing the package If &Alnuth; is not loaded when GAP is started you have to request it explicitly to use it. This is done by calling LoadPackage("alnuth"); in a GAP session. If &Alnuth; had not been loaded already a short banner will be displayed.

LoadPackage("alnuth"); Loading Alnuth 4.0.0 (ALgebraic NUmber THeory and an interface to PARI/GP and OSCAR) by Björn Assmann, Andreas Distler (a.distler@tu-bs.de), and Bettina Eick (http://www.iaa.tu-bs.de/beick). maintained by: Max Horn (https://www.quendi.de/math) and The GAP Team (support@gap-system.org). Homepage: https://gap-packages.github.io/alnuth Report issues at https://github.com/gap-packages/alnuth/issues true gap> ]]> To load a certain version of &Alnuth; you can specify the version number as second argument in the call to LoadPackage. (See in the &GAP; reference manual or type ?LoadPackage within a GAP session).

Once the package is loaded, it is possible to check the correct installation running a short test by calling ReadPackage("Alnuth", "tst/testinstall.tst");.

ReadPackage("Alnuth", "tst/testinstall.g"); Architecture: aarch64-apple-darwin21.4.0-default64-kv8 testing: GAPROOT/pkg/alnuth/tst/ALNUTH.tst 66 ms (33 ms GC) and 11.0MB allocated for GAPROOT/pkg/alnuth/tst/ALNUTH.tst testing: GAPROOT/pkg/alnuth/tst/version.tst 21 ms (21 ms GC) and 29.6KB allocated for GAPROOT/pkg/alnuth/tst/version.tst ----------------------------------- total 87 ms (54 ms GC) and 11.0MB allocated 0 failures in 2 files #I No errors detected while testing gap> ]]> The architecture, timings and memory usage will usually differ; other discrepancies in the output indicate some problem.

If the test suite runs into an error in the first part, which when running outside OSCAR verifies the availability of PARI/GP, check your installation of PARI/GP and consult the last chapter of the documentation of &Alnuth; for more information.

If you find any bugs or have any suggestions or comments, we would very much appreciate it if you would let us know by submitting an issue at the &Alnuth; issue tracker on GitHub https://github.com/gap-packages/alnuth/issues or by writing an mail to support@gap-system.org.

alnuth-4.0.0/doc/intro.xml0000644000000000000000000000337015201210200012340 0ustar00 Introduction A number field is a finite extension of the field of rational numbers. &Alnuth; provides various methods to compute with number fields which are given by a defining polynomial or by generators. For background on number fields we refer to .

Some of the methods provided in this package are written in &GAP; code. The other part of the methods is imported from the Computer Algebra Systems PARI/GP respectively OSCAR , . Hence this package contains some &GAP; functions and an interface to some functions to these computer algebra systems. Therefore one has to have PARI/GP or OSCAR installed to use the full functionality of &Alnuth;.

We note that only a very small part of the functions available in PARI/GP respectively OSCAR are linked to &GAP; and they provides many more methods for computations in number fields.

The main methods included in &Alnuth; are: creating a number field, computing its maximal order, computing its unit group and a presentation of this unit group, computing the elements of a given norm of the number field, determining a presentation for a finitely generated multiplicative subgroup, and factoring polynomials defined over number fields. For background on algorithms for number fields we refer to , and .

The functions provided by &Alnuth; are introduced in the following chapter. Then an example application is outlined. In the final chapter of this manual the installation of the package and configuration of the interface, including hints on the installation of PARI/GP respectively OSCAR, are described.

alnuth-4.0.0/doc/lefttoc.css0000644000000000000000000000047415201210200012637 0ustar00/* leftmenu.css Frank Lübeck */ /* Change default CSS to show section menu on left side */ body { padding-left: 28%; } body.chap0 { padding-left: 2%; } div.ChapSects div.ContSect:hover div.ContSSBlock { left: 15%; } div.ChapSects { left: 1%; width: 25%; } alnuth-4.0.0/doc/manual.bib0000644000000000000000000000522315201210200012415 0ustar00 @manual{GAP4, key = "GAP", organization = "The GAP~Group", title = "{GAP -- Groups, Algorithms, and Programming, Version 4.14.0}", year = 2024, url = {https://www.gap-system.org/}, keywords = "groups; *; gap; manual", } @article{Kant, author = {M. Daberkow and C.Fieker and J. Klüners and M. Pohst and K.Roegner and K. Wildanger}, title = {Kant {V4}}, journal = {J. Symb. Comput.}, volume = 24, year = {1997}, pages = {267 -- 283} } @manual{PARI2, key = "PARI", organization = "{The PARI~Group}", title = "{PARI/GP, version {2.5.0}}", year = 2011, address = "Bordeaux", url = {https://pari.math.u-bordeaux.fr/} } @book{Poh93, author = {Michael E. Pohst}, title = {Computational Algebraic Number Theory}, publisher = {Birkhäuser}, volume = {21}, series = {DMV Seminar}, year = 1993 } @manual{Polycyclic, author = {Bettina Eick and Max Horn and Werner Nickel}, title = {Polycyclic - a {GAP} package}, url = {https://gap-packages.github.io/polycyclic/}, year = {2011}, } @book{PZa89, author = {M. Pohst and H. Zassenhaus}, title = {Algorithmic algebraic number theory}, publisher = {Cambridge University Press}, year = 1989 } @book{Coh93, author = {Henri Cohen}, title = {A course in computational algebraic number theory}, publisher = {Springer-Verlag}, address = {New York, Heidelberg, Berlin}, year = 1993 } @book{Sta79, author = {I. N. Stuart and D. O. Tall}, title = {Algebraic number theory}, publisher = {Chapman and Hall}, year = {1979} } @misc{OSCAR, key = {OSCAR}, organization = {The OSCAR Team}, title = {OSCAR -- Open Source Computer Algebra Research system, Version 1.0.0}, year = {2024}, url = {https://www.oscar-system.org}, } @book{Oscar25, editor = {Decker, Wolfram and Eder, Christian and Fieker, Claus and Horn, Max and Joswig, Michael}, title = {The {C}omputer {A}lgebra {S}ystem {OSCAR}: {A}lgorithms and {E}xamples}, year = {2025}, publisher = {Springer}, series = {Algorithms and Computation in Mathematics}, volume = {32}, edition = {1}, url = {https://doi.org/10.1007/978-3-031-62127-7}, issn = {1431-1550}, doi = {10.1007/978-3-031-62127-7}, }alnuth-4.0.0/doc/manual.css0000644000000000000000000001606615201210200012460 0ustar00/* manual.css Frank Lübeck */ /* This is the default CSS style sheet for GAPDoc HTML manuals. */ /* basic settings, fonts, sizes, colors, ... */ body { position: relative; background: #ffffff; color: #000000; 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"X81488B807F2A1CF1" ], [ "Acknowledgements", "0.0-2", [ 0, 0, 2 ], 48, 2, "acknowledgements", "X82A988D47DFAFCFA" ], [ "Table of Contents", "0.0-3", [ 0, 0, 3 ], 67, 3, "table of contents", "X8537FEB07AF2BEC8" ], [ "\033[1X\033[33X\033[0;-2YIntroduction\033[133X\033[101X", "1", [ 1, 0, 0 ], 1, 4, "introduction", "X7DFB63A97E67C0A1" ], [ "\033[1X\033[33X\033[0;-2YWorking with number fields\033[133X\033[101X", "2", [ 2, 0, 0 ], 1, 5, "working with number fields", "X7DC995CB8412F8E2" ], [ "\033[1X\033[33X\033[0;-2YCreation of number fields\033[133X\033[101X", "2.1", [ 2, 1, 0 ], 10, 5, "creation of number fields", "X86AD0C3F7D38C269" ], [ "\033[1X\033[33X\033[0;-2YMethods for number fields\033[133X\033[101X", "2.2", [ 2, 2, 0 ], 44, 6, "methods for number fields", "X79007A477D53B572" ], [ "\033[1X\033[33X\033[0;-2YPresentations of multiplicative subgroups\033[133\ X\033[101X", "2.3", [ 2, 3, 0 ], 124, 7, "presentations of multiplicative subgroups", "X833A0296798E0463" ], [ 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"X7CC3F0458523CB22" ], [ "\033[2XExampleMatField\033[102X", "2.6-1", [ 2, 6, 1 ], 222, 9, "examplematfield", "X7B4AE27B7BB8A9ED" ], [ "\033[2XPariVersion\033[102X", "4.3-1", [ 4, 3, 1 ], 88, 14, "pariversion", "X84E2E83F8251DC49" ], [ "\033[2XSetAlnuthExternalExecutable\033[102X", "4.3-2", [ 4, 3, 2 ], 140, 15, "setalnuthexternalexecutable", "X7F2835FB7C5069DD" ], [ "\033[2XSetAlnuthExternalExecutablePermanently\033[102X", "4.3-3", [ 4, 3, 3 ], 152, 15, "setalnuthexternalexecutablepermanently", "X81CD0C70812FC6F3" ], [ "\033[2XRestoreAlnuthExternalExecutablePermanently\033[102X", "4.3-4", [ 4, 3, 4 ], 161, 15, "restorealnuthexternalexecutablepermanently", "X7F648B427EB19552" ] ] ); alnuth-4.0.0/doc/nocolorprompt.css0000644000000000000000000000031315201210200014104 0ustar00 /* colors for ColorPrompt like examples */ span.GAPprompt { color: #000000; font-weight: normal; } span.GAPbrkprompt { color: #000000; font-weight: normal; } span.GAPinput { color: #000000; } alnuth-4.0.0/doc/ragged.css0000644000000000000000000000023115201210200012417 0ustar00/* times.css Frank Lübeck */ /* Change default CSS to use Times font. */ body { text-align: left; } alnuth-4.0.0/doc/rainbow.js0000644000000000000000000000533615201210200012466 0ustar00 function randchar(str) { var i = Math.floor(Math.random() * str.length); while (i == str.length) i = Math.floor(Math.random() * str.length); return str[i]; } hexdigits = "0123456789abcdef"; function randlight() { return randchar("cdef")+randchar(hexdigits)+ randchar("cdef")+randchar(hexdigits)+ randchar("cdef")+randchar(hexdigits) } function randdark() { return randchar("012345789")+randchar(hexdigits)+ randchar("012345789")+randchar(hexdigits)+ randchar("102345789")+randchar(hexdigits) } document.write('\n'); alnuth-4.0.0/doc/times.css0000644000000000000000000000026115201210200012312 0ustar00/* times.css Frank Lübeck */ /* Change default CSS to use Times font. */ body { font-family: Times,Times New Roman,serif; } alnuth-4.0.0/doc/title.xml0000644000000000000000000000434315201210200012327 0ustar00 Alnuth ALgebraic NUmber THeory
 and an interface to
 PARI/GP and OSCAR &VERSION; Note: PARI/GP is not part of this package. It can be obtained from https://pari.math.u-bordeaux.fr/. If you use GAP via OSCAR, then OSCAR will automatically be used instead of PARI/GP. GAP code by Björn Assmann, Andreas Distler and Bettina Eick PARI/GP code by Bill Allombert OSCAR code by Claus Fieker and Max Horn &RELEASEDATE; This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 2 of the license, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this program. If not, see https://www.gnu.org/licenses/

To begin with we are very grateful for all the feedback by users of former versions of &Alnuth;.

We thank Bill Allombert who wrote the GP code for the interface to PARI/GP and who was extremely helpful in the transition from KANT to PARI/GP.

For feedback on the development version, including a code patch, we are much obliged to Max Horn.

The second author acknowledges the financial support at CAUL within the projects PTDC/MAT/101993/2008 and ISFL-1-143, financed by FEDER and FCT, in the development of Version 3 of &Alnuth;.

Support for using OSCAR instead of PARI/GP was added by Claus Fieker and Max Horn.

alnuth-4.0.0/doc/toggless.css0000644000000000000000000000167215201210200013027 0ustar00/* toggless.css Frank Lübeck */ /* Using javascript we change all div.ContSect to div.ContSectOpen or div.ContSectClosed. This way the config for div.ContSect in manual.css is no longer relevant. Here we add the CSS for the new elements. */ /* This layout is based on an idea by Burkhard Höfling. */ div.ContSectClosed { text-align: left; margin-left: 1em; } div.ContSectOpen { text-align: left; margin-left: 1em; } div.ContSectOpen div.ContSSBlock { display: block; text-align: left; margin-left: 1em; } div.ContSectOpen div.ContSSBlock a { display: block; width: 100%; margin-left: 1em; } span.tocline a:hover { display: inline; background: #eeeeee; } span.ContSS a:hover { display: inline; background: #eeeeee; } span.toctoggle { font-size: 80%; display: inline-block; width: 1.2em; } span.toctoggle:hover { background-color: #aaaaaa; } alnuth-4.0.0/doc/toggless.js0000644000000000000000000000420515201210200012646 0ustar00/* toggless.js Frank Lübeck */ /* this file contains two functions: mergeSideTOCHooks: this changes div.ContSect elements to the class ContSectClosed and includes a hook to toggle between ContSectClosed and ContSectOpen. openclosetoc: this function does the toggling, the rest is done by CSS */ closedTOCMarker = "▶ "; openTOCMarker = "▼ "; noTOCMarker = " "; /* merge hooks into side toc for opening/closing subsections with openclosetoc */ function mergeSideTOCHooks() { var hlist = document.getElementsByTagName("div"); for (var i = 0; i < hlist.length; i++) { if (hlist[i].className == "ContSect") { var chlds = hlist[i].childNodes; var el = document.createElement("span"); var oncl = document.createAttribute("class"); oncl.nodeValue = "toctoggle"; el.setAttributeNode(oncl); var cont; if (chlds.length > 2) { var oncl = document.createAttribute("onclick"); oncl.nodeValue = "openclosetoc(event)"; el.setAttributeNode(oncl); cont = document.createTextNode(closedTOCMarker); } else { cont = document.createTextNode(noTOCMarker); } el.appendChild(cont); hlist[i].firstChild.insertBefore(el, hlist[i].firstChild.firstChild); hlist[i].className = "ContSectClosed"; } } } function openclosetoc (event) { /* first two steps to make it work in most browsers */ var evt=window.event || event; if (!evt.target) evt.target=evt.srcElement; var markClosed = document.createTextNode(closedTOCMarker); var markOpen = document.createTextNode(openTOCMarker); var par = evt.target.parentNode.parentNode; if (par.className == "ContSectOpen") { par.className = "ContSectClosed"; evt.target.replaceChild(markClosed, evt.target.firstChild); } else if (par.className == "ContSectClosed") { par.className = "ContSectOpen"; evt.target.replaceChild(markOpen, evt.target.firstChild); } } /* adjust jscontent which is called onload */ jscontentfuncs.push(mergeSideTOCHooks); alnuth-4.0.0/exam/0000755000000000000000000000000015201210200010645 5ustar00alnuth-4.0.0/exam/fields.gi0000644000000000000000000000207015201210200012433 0ustar00############################################################################# ## #W fields.gi Alnuth - ALgebraic NUmber THeory Bjoern Assmann ## ############################################################################# ## ## ExampleMatField provides somes examples of fields which are generated ## by matrices in GL(d,Z) or GL(d,Q) ## BindGlobal( "ExampleMatField", function(n) if n = 1 then return FieldByMatricesNC(ExamUnimod(1){[1..4]}); elif n = 2 then return FieldByMatricesNC(ExamUnimod(2){[1..4]}); elif n = 3 then return FieldByMatricesNC(ExamUnimod(3){[1..4]}); elif n = 4 then return FieldByMatricesNC(ExamUnimod(1)); elif n = 5 then return FieldByMatricesNC(ExamUnimod(2)); elif n = 6 then return FieldByMatricesNC(ExamUnimod(3)); elif n = 7 then return FieldByMatricesNC(ExamRationals(1)); elif n = 8 then return FieldByMatricesNC(ExamRationals(2)); elif n = 9 then return FieldByMatricesNC(ExamRationals(3)); else return fail; fi; end ); alnuth-4.0.0/exam/rationals.gi0000644000000000000000000003420415201210200013165 0ustar00############################################################################# ## #W rationals.gi Alnuth - ALgebraic NUmber THeory Bjoern Assmann ## ############################################################################# ## ## ExamRationals contains 3 lists of ## matrices in GL(d,Q) which generated a free-ablian group ## such that the associated algebra over Q is simple over ## BindGlobal( "ExamRationals", function(n) local matrixlist; matrixlist:= [ [ [ [ -260689/5, -2887416/25, 1206744/125, 1448433/25 ], [ 236502/5, 2619208/25, -1081647/125, -1314054/25 ], [ -420105/4, -232680, 19657, 233415/2 ], [ 222054/5, 2458876/25, -1001784/125, -1233788/25 ] ], [ [ -151553/25, -2649108/25, -363216/25, 1720341/25 ], [ 654381/100, 4794304/25, 719523/25, -6309891/50 ], [ -9870, 0, 5545, -14805/4 ], [ 181608/25, 7064288/25, 1093596/25, -4671026/25 ] ], [ [ -116112/25, -29429764/75, -7979888/125, 6537587/25 ], [ 892899/100, 18795319/25, 15287769/125, -25050837/50 ], [ 1120, 278320/3, 15086, -247275/4 ], [ 335832/25, 28253568/25, 22980468/125, -18828407/25 ] ], [ [ 62523793214089/25, 5339768524608, 2430717299136/125, -67628119554576/25 ], [ -124713368896032/25, -10836657663551, -8719832238768/125, 138007269305388/25 ], [ -613170403020, -1330061172720, -8251382159, 677232982440 ], [ -189793628149104/25, -16575734835072, -15023590053696/125, 211434513536761/25 ] ], [ [ 206728/25, 2649108/25, 363216/25, -1720341/25 ], [ -654381/100, -4739129/25, -719523/25, 6309891/50 ], [ 9870, 0, -3338, 14805/4 ], [ -181608/25, -7064288/25, -1093596/25, 4726201/25 ] ], [ [ 62523793214089/25, 5339768524608, 2430717299136/125, -67628119554576/25 ], [ -124713368896032/25, -10836657663551, -8719832238768/125, 138007269305388/25 ], [ -613170403020, -1330061172720, -8251382159, 677232982440 ], [ -189793628149104/25, -16575734835072, -15023590053696/125, 211434513536761/25 ] ], [ [ 262548367/25, 22422624, 10207008/125, -283982328/25 ], [ -523693296/25, -45505025, -36616104/125, 579516714/25 ], [ -2574810, -5585160, -34649, 2843820 ], [ -796976712/25, -69604416, -63086688/125, 887850583/25 ] ], [ [ 62523793214089/25, 5339768524608, 2430717299136/125, -67628119554576/25 ], [ -124713368896032/25, -10836657663551, -8719832238768/125, 138007269305388/25 ], [ -613170403020, -1330061172720, -8251382159, 677232982440 ], [ -189793628149104/25, -16575734835072, -15023590053696/125, 211434513536761/25 ] ], [ [ -61106002910039/25, -5339768524608, -2430717299136/125, 67628119554576/25 ], [ 124713368896032/25, 10893369275713, 8719832238768/125, -138007269305388/25 ], [ 613170403020, 1330061172720, 64962994321, -677232982440 ], [ 189793628149104/25, 16575734835072, 15023590053696/125, -210016723232711/25 ] ], [ [ -29146612742563984/25, -38742944058068404/15, 27229938402618896/125, 32388330250096051/25 ], [ 104624553390591843/100, 11589291802956299/5, -24436109013984423/125, -58130504103186501/50 ], [ -2371390466258800, -15760776275028400/3, 443089804313262, 10540556233557525/4 ], [ 24257889033477624/25, 10748213265061888/5, -22662680336012556/125, -26955877378349111/25 ] ], [ [ -29146612742563984/25, -38742944058068404/15, 27229938402618896/125, 32388330250096051/25 ], [ 104624553390591843/100, 11589291802956299/5, -24436109013984423/125, -58130504103186501/50 ], [ -2371390466258800, -15760776275028400/3, 443089804313262, 10540556233557525/4 ], [ 24257889033477624/25, 10748213265061888/5, -22662680336012556/125, -26955877378349111/25 ] ], [ [ -61106002910039/25, -5339768524608, -2430717299136/125, 67628119554576/25 ], [ 124713368896032/25, 10893369275713, 8719832238768/125, -138007269305388/25 ], [ 613170403020, 1330061172720, 64962994321, -677232982440 ], [ 189793628149104/25, 16575734835072, 15023590053696/125, -210016723232711/25 ] ], [ [ -61106002910039/25, -5339768524608, -2430717299136/125, 67628119554576/25 ], [ 124713368896032/25, 10893369275713, 8719832238768/125, -138007269305388/25 ], [ 613170403020, 1330061172720, 64962994321, -677232982440 ], [ 189793628149104/25, 16575734835072, 15023590053696/125, -210016723232711/25 ] ], [ [ -61106002910039/25, -5339768524608, -2430717299136/125, 67628119554576/25 ], [ 124713368896032/25, 10893369275713, 8719832238768/125, -138007269305388/25 ], [ 613170403020, 1330061172720, 64962994321, -677232982440 ], [ 189793628149104/25, 16575734835072, 15023590053696/125, -210016723232711/25 ] ], [ [ -151553/25, -2649108/25, -363216/25, 1720341/25 ], [ 654381/100, 4794304/25, 719523/25, -6309891/50 ], [ -9870, 0, 5545, -14805/4 ], [ 181608/25, 7064288/25, 1093596/25, -4671026/25 ] ], [ [ -64265363669/25, -5862628307256/25, -4777859393832/125, 3908821093569/25 ], [ 122902951122/25, 11230823634328/25, 9153133403841/125, -7488059052222/25 ], [ 2425896375/4, 55411113240, 9031991977, -73889742465/2 ], [ 184645147134/25, 16881759134716/25, 13758822703752/125, -11255808020684/25 ] ], [ [ 206728/25, 2649108/25, 363216/25, -1720341/25 ], [ -654381/100, -4739129/25, -719523/25, 6309891/50 ], [ 9870, 0, -3338, 14805/4 ], [ -181608/25, -7064288/25, -1093596/25, 4726201/25 ] ], [ [ 262548367/25, 22422624, 10207008/125, -283982328/25 ], [ 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] ] , [ [ [ -6264153250196389416903240923/229, 32298639086108555190545138052/229, 6250050153899910196592931288/229, -17062528604518090575374454108/229 ], [ -31828949487280183133780637174/229, 164113442139592040994066347761/229, 31757289883546699739121527862/229, -86696851016768019463271091168/229 ], [ 10706305822230402790227171324/229, -55202849273663798845585812684/229, -10682201676632474178149363807/229, 29162225450789804731984637796/229 ], [ -31630873935512564569306687014/229, 163092143569330601229006182316/229, 31559660278496695307183104530/229, -86157325619974009272661504835/229 ] ], [ [ -32922740572311041966823464707818129930409331/229, 169753145077879383397309722705981485900163876/229, 32848618410529519585754676836021522433280506/229, -89676158982313211602360836468734114418152304/229 ], [ -167284580182655189842874359703358730654824807/229, 862537052365596961509145685227392451967278339/229, 166907956168362736093277019723482684077697700/229, -455655827764409867117421029606966266278026982/229 ], [ 56269525184758861487331252988208249131583094/229, -290131644756973897407392677387760533570749882/229, -56142840140421394598033623641159798843012641/229, 153268980607634202422715921838991026204863618/229 ], [ -166243547221918122310167837625294921493119821/229, 857169375916344158272836981518015370204924276/229, 165869266986187060467986580374823273478628004/229, -452820224297929303195372763004426259134893423/229 ] ], [ [ 15249919019557/229, -10501382066448/229, -688215138912/229, 7314116210352/229 ], [ 6402597223176/229, -20493339577979/229, -7017597985608/229, 16978413905712/229 ], [ -7519116464496/229, 32192361338256/229, 11640450259093/229, -26964854857584/229 ], [ 3766879669896/229, -23145504884544/229, -8767421583480/229, 19576888669525/229 ] ], [ [ 80555126206508585287903316821/229, -183335834990114612065968727812/229, -57125319139708944868540766808/229, 149195252972813686479801033948/229 ], [ 27897137130063091461241343094/229, -63491240972825988770606150903/229, -19783134068385556918206404982/229, 51667977288833930017011925728/229 ], [ -29769638463282968475266172924/229, 67752876595690071254903155404/229, 21111010285418575004268784345/229, -55136016174223925477850277476/229 ], [ 12329249636446841813204697894/229, -28060204028541594293313254316/229, -8743234023736905752902049490/229, 22834866073687996604786287933/229 ] ], [ [ 31249965280/229, -21519315909/229, -1410282846/229, 14988005991/229 ], [ 26240262741/458, -41994724658/229, -28760768253/458, 34791977871/229 ], [ -15408090243/229, 65968230573/229, 23853481843/229, -55256082147/229 ], [ 15438096261/458, -47429512452/229, -35932206555/458, 40116743599/229 ] ], [ [ 15327076463725/229, -34882860138228/229, -10869082372854/229, 28387009458072/229 ], [ 5307942503781/229, -12080402052989/229, -3764116093752/229, 9830805308370/229 ], [ -5664223486098/229, 12891306835806/229, 4016789208679/229, -10490706552294/229 ], [ 2345870499615/229, -5339047456092/229, -1663592552064/229, 4344819846889/229 ] ], [ [ 1420434424899105493/229, -3232774170942360564/229, -1007294925359913336/229, 2630770794283255596/229 ], [ 491912256883196718/229, -1119545690071021655/229, -348837448172916654/229, 911065217849276976/229 ], [ -524930209668704268/229, 1194691422286793148/229, 372251986514043241/229, -972217400820933492/229 ], [ 217402559487699678/229, -494787627432359532/229, -154170084225283290/229, 402648861567049357/229 ] ] ] ]; return matrixlist[n]; end ); alnuth-4.0.0/exam/unimod.gi0000644000000000000000000001654515201210200012474 0ustar00############################################################################# ## #W unithom.gi Alnuth - ALgebraic NUmber THeory Bjoern Assmann ## ############################################################################# ## ## ExamUnimod contains 3 lists of matrices in GL(d,Z). Each of them ## generates a torsion-free abelian group with semisimple matrix algebra. ## ## Remark: These exemples were generated as normalsubgroupsgenerators ## of the kernel of the p-congruence homomorphism applied to ## MatExamples. The order is changes ## MatExamples(2) corresponds to ExamUnimod[1] ## MatExamples(3) corresponds to ExamUnimod[2] ## MatExamples(4) corresponds to ExamUnimod[3] ############################################################################# ## #F ExamUnimod(n) ## BindGlobal( "ExamUnimod", function(n) local matrixlist; matrixlist:= [ [ [ [ 2093340575128, 6213435078921, 17560570361343, -28693642181391 ], [ -3625838403333, -10761656507462, -30415567443933, 49698063067818 ], [ 1293798337794, 3839901865707, 10852879181962, -17733140676042 ] , [ -1314714587787, -3902311263375, -11028829922394, 18020890988656 ] ], [ [ 398946457, -197870628, -271469520, 913773168 ], [ 690995412, -342721799, -470198736, 1582701120 ], [ 246562392, -122290656, -167777219, 564742596 ], [ 329632680, -163492104, -224303736, 755012245 ] ], [ [ 57641556673, -51250063536, -73214376480, 161071628256 ], [ 21964312944, 28355806081, 43928625888, 0 ], [ -14642875296, 43928625888, 64962994321, -80535814128 ], [ 0, 29285750592, 43928625888, -37537132751 ] ], [ [ 13, 0, -21, 0 ], [ -42, 97, 105, -231 ], [ -21, 0, 34, 0 ], [ -21, 21, 42, -50 ] ], [ [ 6113341760402965, -3032143586011050, -4159967272068153, 14002438585824810 ], [ 10588511869480164, -5251666322974043, -7205040811308855, 24252552936374367 ], [ 3778184141734557, -1873864241780610, -2570850209123252, 8653736311352880 ], [ 5051133104082267, -2505235179386847, -3437063756709534, 11569395035183716 ] ], [ [ -929944511, 51250063536, 73214376480, -161071628256 ], [ -21964312944, 28355806081, -43928625888, 0 ], [ 14642875296, -43928625888, -8251382159, 80535814128 ], [ 0, -29285750592, -43928625888, 94248744913 ] ], [ [ 66665305, 197870628, 559233360, -913773168 ], [ -115467732, -342721799, -968620464, 1582701120 ], [ 41201448, 122290656, 345625141, -564742596 ], [ -41868840, -124271736, -351223944, 573891037 ] ], [ [ -929944511, 51250063536, 73214376480, -161071628256 ], [ -21964312944, 28355806081, -43928625888, 0 ], [ 14642875296, -43928625888, -8251382159, 80535814128 ], [ 0, -29285750592, -43928625888, 94248744913 ] ], [ [ -929944511, 51250063536, 73214376480, -161071628256 ], [ -21964312944, 28355806081, -43928625888, 0 ], [ 14642875296, -43928625888, -8251382159, 80535814128 ], [ 0, -29285750592, -43928625888, 94248744913 ] ], [ [ 34, 0, 21, 0 ], [ 42, -50, -105, 231 ], [ 21, 0, 13, 0 ], [ 21, -21, -42, 97 ] ], [ [ 3385900582, -1680337008, -2305472631, 7756555653 ], [ 5861155020, -2905320200, -3985748151, 13422579324 ], [ 2090409006, -1035207621, -1420053053, 4785959871 ], [ 2795550429, -1385267583, -1900360644, 6401474647 ] ], [ [ 13, 0, -21, 0 ], [ -42, 97, 105, -231 ], [ -21, 0, 34, 0 ], [ -21, 21, 42, -50 ] ], [ [ 242047, -215208, -307440, 676368 ], [ 92232, 119071, 184464, 0 ], [ -61488, 184464, 272791, -338184 ], [ 0, 122976, 184464, -157625 ] ] ] , [ [ [ -17555, -25515, -19929, 32571 ], [ 30387, 44164, 34503, -56364 ], [ 28392, 41265, 32236, -52668 ], [ 19362, 28140, 21987, -35909 ] ], [ [ 1351, -780, 0, 0 ], [ -2340, 1351, 0, 0 ], [ 5460, 6240, 5251, -8580 ], [ 780, 2340, 1560, -2549 ] ], [ [ 5044200094, 3284045655, 3464193024, -5660403408 ], [ -8736810795, -5688133874, -6000158304, 9804106224 ], [ 3117487071, 2029651800, 2140989019, -3498321651 ], [ -2217936039, -1443995685, -1523206542, 2488880785 ] ], [ [ 1351, 780, 0, 0 ], [ 2340, 1351, 0, 0 ], [ -5460, -6240, -2549, 8580 ], [ -780, -2340, -1560, 5251 ] ], [ [ 445, 285, 303, -495 ], [ -771, -494, -525, 858 ], [ 288, 195, 202, -330 ], [ -192, -120, -129, 211 ] ], [ [ 5044200094, 3284045655, 3464193024, -5660403408 ], [ -8736810795, -5688133874, -6000158304, 9804106224 ], [ 3117487071, 2029651800, 2140989019, -3498321651 ], [ -2217936039, -1443995685, -1523206542, 2488880785 ] ], [ [ -64949609, -94398330, -73739913, 120489633 ], [ 112494768, 163500796, 127720299, -208690878 ], [ 105089865, 152738475, 119313055, -194954199 ], [ 71703495, 104214405, 81408309, -133017998 ] ], [ [ -3051232082, -4434686265, -3464193024, 5660403408 ], [ 5284888965, 7681101886, 6000158304, -9804106224 ], [ 4936997199, 7175473080, 5605182043, -9158725059 ], [ 3368576649, 4895917515, 3824487762, -6249115823 ] ], [ [ -3051232082, -4434686265, -3464193024, 5660403408 ], [ 5284888965, 7681101886, 6000158304, -9804106224 ], [ 4936997199, 7175473080, 5605182043, -9158725059 ], [ 3368576649, 4895917515, 3824487762, -6249115823 ] ], [ [ 1351, -780, 0, 0 ], [ -2340, 1351, 0, 0 ], [ 5460, 6240, 5251, -8580 ], [ 780, 2340, 1560, -2549 ] ], [ [ 181150, 193440, 73353, -246906 ], [ 313755, 335044, 127047, -427647 ], [ -293103, -312990, -118685, 399498 ], [ 164082, 175215, 66441, -223643 ] ], [ [ 1351, -780, 0, 0 ], [ -2340, 1351, 0, 0 ], [ 5460, 6240, 5251, -8580 ], [ 780, 2340, 1560, -2549 ] ], [ [ 220027, 143250, 151107, -246906 ], [ -381093, -248111, -261723, 427647 ], [ 135981, 88530, 93388, -152592 ], [ -96744, -62985, -66441, 108562 ] ] ] , [ [ [ -492568055, -715902540, -559233360, 913773168 ], [ 853152732, 1239979321, 968620464, -1582701120 ], [ 796991748, 1158354480, 904858501, -1478515764 ], [ 543797628, 790360020, 617396520, -1008810083 ] ], [ [ -348686135, -530151780, -271469520, 913773168 ], [ -603941868, -918249479, -470198736, 1582701120 ], [ -215499732, -327651600, -167777219, 564742596 ], [ -793008492, -1205711460, -617396520, 2078172517 ] ], [ [ -151935722, -225045135, -150030090, 330066198 ], [ 45009027, 58106404, 90018054, 0 ], [ 105021063, 150030090, 133121449, -165033099 ], [ -45009027, -75015045, 0, 193133485 ] ], [ [ 13, 0, -21, 0 ], [ 147, 181, 105, -231 ], [ -21, 0, 34, 0 ], [ 84, 105, 63, -134 ] ], [ [ 670415977, 715902540, 271469520, -913773168 ], [ 1161194148, 1239979321, 470198736, -1582701120 ], [ -1084755588, -1158354480, -439246739, 1478515764 ], [ 607257732, 648459180, 245895000, -827688875 ] ], [ [ 814297897, 530151780, 559233360, -913773168 ], [ -1410405012, -918249479, -968620464, 1582701120 ], [ 503263572, 327651600, 345625141, -564742596 ], [ -358046868, -233107740, -245895000, 401786125 ] ], [ [ 141577573978831, 92174617804950, 97230886083807, -158872797828558 ], [ -245219551343241, -159651121205951, -168408834761919, 275175757779087 ], [ 87499752763389, 56967046703190, 60091992355948, -98188788945456 ], [ -62251695137616, -40529202794985, -42752445237981, 69856409444878 ] ] ] ]; return matrixlist[n]; end ); alnuth-4.0.0/gap/0000755000000000000000000000000015201210200010462 5ustar00alnuth-4.0.0/gap/factors.gd0000644000000000000000000000274315201210200012445 0ustar00############################################################################# ## #W factors.gd Alnuth - ALgebraic NUmber THeory Andreas Distler ## ############################################################################# ## #F FactorsPolynomialAlnuth, function( ) ## ## Factorizes the polynomial defined over an algebraic extension of ## the rationals using PARI/GP ## ## As a method of 'Factors' ? AD ## DeclareGlobalFunction("FactorsPolynomialAlnuth"); DeclareObsoleteSynonym( "FactorsPolynomialPari", "FactorsPolynomialAlnuth" ); ############################################################################# ## #F FactorsPolynomialAlgExt, function( , ) ## ## Factorizes the rational polynomial over the field , a proper ## algebraic extension of the rationals, using PARI/GP ## DeclareGlobalFunction( "FactorsPolynomialAlgExt" ); ############################################################################# ## #M IrrFacsAlgExtPol() . . . . . lists of irreducible factors of rational ## polynomial over algebraic extensions, initialize default ## DeclareAttribute( "IrrFacsAlgExtPol", IsUnivariatePolynomial, "mutable" ); ############################################################################# ## #F StoreFactorsAlgExtPol( , , ) . store factors list ## DeclareGlobalFunction( "StoreFactorsAlgExtPol" ); ############################################################################# ## #E alnuth-4.0.0/gap/factors.gi0000644000000000000000000000660615201210200012454 0ustar00############################################################################ ## #W factors.gi Alnuth - ALgebraic NUmber THeory Andreas Distler ## ############################################################################# ## #M IrrFacsAlgExtPol() . . . . . lists of irreducible factors of rational ## polynomial over algebraic extensions, initialize default ## InstallOtherMethod(IrrFacsAlgExtPol,true,[IsPolynomial],0,f -> []); ############################################################################# ## #F StoreFactorsAlgExtPol( , , ) . store factors list ## InstallGlobalFunction(StoreFactorsAlgExtPol,function(R,f,fact) local irf; irf:=IrrFacsAlgExtPol(f); if not ForAny(irf,i->i[1]=R) then Add(irf,[R,fact]); fi; end); ############################################################################# ## #F FactorsPolynomialAlgExt, function( , ) ## ## Factorizes the rational polynomial over the field , a proper ## algebraic extension of the rationals, using PARI/GP ## InstallGlobalFunction( FactorsPolynomialAlgExt, function( H, poly ) local faktoren, irf, i; if not ForAll( CoefficientsOfUnivariatePolynomial( poly ), IsRat ) then Error( "polynomial has to be defined over the Rationals" ); fi; if H = Rationals then return Factors( poly ); fi; irf := IrrFacsAlgExtPol( poly ); i := PositionProperty( irf, k -> k[1] = H ); if i <> fail then return irf[i][2]; fi; faktoren := FactorsPolynomialAlnuth( AlgExtEmbeddedPol( H, poly )); StoreFactorsAlgExtPol( H, poly, faktoren ); return faktoren; end ); ############################################################################# ## #F FactorsPolynomialAlnuth, function( ) ## ## Factorizes the polynomial defined over an algebraic extension of ## the rationals using PARI/GP ## ## As a method of 'Factors' ? AD ## InstallGlobalFunction(FactorsPolynomialAlnuth, function( poly ) local faktoren, fak, coeff, c, lcoeff, irf, i, coeffs, H; H := CoefficientsRing( DefaultRing( poly )); irf := IrrFacsAlgExtPol( poly ); i := PositionProperty( irf, k -> k[1] = H ); if i <> fail then return irf[i][2]; fi; if DegreeOfLaurentPolynomial( poly ) < 2 then faktoren := [ poly ]; StoreFactorsPol( H, poly, faktoren ); return faktoren; fi; faktoren := [ ]; lcoeff := LeadingCoefficient( poly ); coeffs := CoefficientsOfUnivariatePolynomial( poly / lcoeff ); coeffs := List( Reversed( coeffs ), ExtRepOfObj ); for fak in AL_FUNCS.PolynomialFactorsDescription( H, coeffs ) do coeff := [ ]; for c in Reversed( fak ) do if ( c in Rationals ) then Add( coeff, c ); else Add( coeff, LinearCombination( EquationOrderBasis(H), c ) ); fi; od; Add( faktoren, UnivariatePolynomial( H, One(H)*coeff ) ); od; faktoren[1] := lcoeff * faktoren[1]; StoreFactorsPol( H, poly, faktoren ); return faktoren; end ); ############################################################################# ## #E alnuth-4.0.0/gap/field.gd0000644000000000000000000000426715201210200012072 0ustar00############################################################################# ## #W field.gd Alnuth - ALgebraic NUmber THeory Bettina Eick #W Bjoern Assmann ## DeclareInfoClass( "InfoAlnuth" ); DeclareRepresentation( "IsBasisOfMatrixField", IsBasis and IsAttributeStoringRep, [] ); DeclareOperation( "ExponentsOfUnits", [IsNumberField, IsCollection] ); DeclareOperation( "IsPrimitiveElementOfNumberField", [ IsNumberField, IsObject ] ); DeclareOperation( "RelationLattice", [IsNumberField, IsCollection] ); DeclareProperty( "IsUnitGroup", IsGroup ); InstallTrueMethod( IsGroup, IsUnitGroup ); DeclareProperty( "IsUnitGroupIsomorphism", IsMapping); DeclareProperty( "IsNumberFieldByMatrices", IsNumberField ); InstallTrueMethod( IsNumberField, IsNumberFieldByMatrices ); DeclareProperty( "IsMultGroupByFieldElemsIsomorphism", IsMapping); DeclareAttribute( "IntegerDefiningPolynomial", IsNumberField ); DeclareAttribute( "IntegerPrimitiveElement", IsNumberField ); DeclareAttribute( "EquationOrderBasis", IsNumberField ); DeclareAttribute( "MaximalOrderBasis", IsNumberField ); DeclareAttribute( "UnitGroup", IsNumberField ); DeclareAttribute( "DefiningPolynomial", IsNumberFieldByMatrices ); DeclareAttribute( "FieldOfUnitGroup", IsGroup ); DeclareGlobalFunction( "FieldOfPolynomial" ); DeclareGlobalFunction( "FieldByMatricesNC" ); DeclareGlobalFunction( "FieldByMatrixBasisNC" ); DeclareGlobalFunction( "FieldByPolynomialNC" ); DeclareGlobalFunction( "FieldByMatrices" ); DeclareGlobalFunction( "FieldByMatrixBasis" ); DeclareGlobalFunction( "FieldByPolynomial" ); DeclareGlobalFunction( "IntersectionOfUnitSubgroups" ); DeclareGlobalFunction( "IntersectionOfTFUnitsByCosets" ); DeclareGlobalFunction( "NormCosetsOfNumberField" ); DeclareGlobalFunction( "IsUnitOfNumberField" ); DeclareGlobalFunction( "RelationLatticeOfTFUnits"); DeclareGlobalFunction( "RelationLatticeModUnits"); DeclareGlobalFunction( "RelationLatticeTF"); DeclareGlobalFunction( "RelationLatticeOfUnits"); DeclareGlobalFunction( "PcpPresentationMultGroupByFieldEl"); DeclareGlobalFunction( "PcpPresentationOfMultiplicativeSubgroup"); alnuth-4.0.0/gap/field.gi0000644000000000000000000001776115201210200012102 0ustar00############################################################################ ## #W field.gi Alnuth - ALgebraic NUmber THeory Bettina Eick #W Bjoern Assmann #W Andreas Distler ## ############################################################################# ## #F ListElmPower( range, elm ) #M EquationOrderBasis( F ) ## BindGlobal( "ListElmPower", function( range, elm ) local list, i; if not IsRange( range ) then Error( " has to be a range" ); fi; # trivial cases if IsEmpty( range ) then return [ ]; elif Length( range ) = 1 then return [ elm ^ range[1] ]; fi; list := [ elm ^ range[1] ]; elm := elm ^ (range[2] - range[1]); for i in [ 1..Length( range ) - 1 ] do Add( list, list[i] * elm ); od; return list; end ); InstallMethod( EquationOrderBasis, "for number field", true, [IsNumberField], 0, function( F ) return RelativeBasisNC( Basis( F ), ListElmPower( [ 0..DegreeOverPrimeField( F )-1 ], IntegerPrimitiveElement( F ) ) ) ; end ); InstallMethod( IsPrimitiveElementOfNumberField, "for number field and algebraic element", true, [ IsNumberField, IsObject ], 0, function( F, elm ) local d, g; if not elm in F then Info( InfoAlnuth, 1, "Element does not lie in the field." ); return false; fi; d := DegreeOverPrimeField( F ); g := MinimalPolynomial( Rationals, elm ); return Degree(g) = d; end ); InstallOtherMethod( EquationOrderBasis, "for number field and primitive element", true, [IsNumberField, IsObject ], 0, function( F , elm ) if not IsPrimitiveElementOfNumberField( F, elm ) then return fail; fi; return RelativeBasisNC(Basis( F ), ListElmPower([0..DegreeOverPrimeField(F)-1],elm)); end ); ############################################################################# ## #M MaximalOrderBasis( F ) ## InstallMethod(MaximalOrderBasis, "for number field", true,[IsNumberField], 0, function( F ) local e, T, b, B; if DegreeOverPrimeField(F)=1 then return EquationOrderBasis(F); fi; e := EquationOrderBasis(F); T := AL_FUNCS.MaximalOrderDescription(F); b := List( T, x -> LinearCombination( e, x ) ); B := Objectify(NewType(FamilyObj(F), IsFiniteBasisDefault and IsRelativeBasisDefaultRep), rec()); SetUnderlyingLeftModule( B, F ); SetBasisVectors( B, b ); B!.basis := e; B!.basechangeMatrix := Immutable( T^-1 ); return B; end ); ############################################################################# ## #F IsIntegerOfNumberField( F, k ) ## BindGlobal( "IsIntegerOfNumberField", function( F, k ) local c; c := Coefficients( MaximalOrderBasis(F), k ); return ForAll( c, IsInt ); end ); ############################################################################# ## #M UnitGroup( F ) ## BindGlobal( "AddNaturalHomomorphismOfUnitGroup", function( G ) local gens, rels, H, nat; # the generators of G are independent and the first one is torsion gens := GeneratorsOfGroup(G); rels := List( gens, x -> 0 ); rels[1] := Order( gens[1] ); H := AbelianPcpGroup( Length(rels), rels ); nat := GroupHomomorphismByImagesNC( G, H, gens, AsList(Pcp(H)) ); # add infos SetIsBijective( nat, true ); SetIsUnitGroupIsomorphism( nat, true ); SetIsomorphismPcpGroup( G, nat ); end ); BindGlobal( "AddUnitGroupOfNumberField", function( F, units ) local gens, G; # check if units are known if HasUnitGroup(F) then gens := GeneratorsOfGroup(UnitGroup(F)); if not gens = units then Error("wrong units"); fi; fi; # otherwise add them G := GroupByGenerators( units ); SetIsUnitGroup( G, true ); SetFieldOfUnitGroup( G, F ); AddNaturalHomomorphismOfUnitGroup( G ); SetUnitGroup( F, G ); end ); BindGlobal( "UnitGroupOfNumberField", function( F ) local eqn, uni, gen, G, r, H, nat; # determine generators eqn := EquationOrderBasis(F); uni := AL_FUNCS.UnitGroupDescription(F); if uni=[-1] then G:=GroupByGenerators([-1*eqn[1]]); else gen := List( uni, x -> LinearCombination( eqn, x ) ); G := GroupByGenerators(gen); fi; # add info SetIsUnitGroup( G, true ); SetFieldOfUnitGroup( G, F ); AddNaturalHomomorphismOfUnitGroup( G ); # return return G; end ); InstallMethod( UnitGroup, "for number field", true, [IsNumberField], 0, function( F ) return UnitGroupOfNumberField( F ); end); ############################################################################# ## #F IsUnitOfNumberField( F, k ) ## InstallGlobalFunction( IsUnitOfNumberField, function( F, k ) if not IsIntegerOfNumberField( F, k ) then return false; fi; return AbsInt(Norm( F, k )) = 1; end ); ############################################################################# ## #F AL_ExponentsTrivialUnits( unit, one ) #F ExponentsOfUnitsOfNumberField( F, elms ) #M ExponentsOfUnits( F, elms ) ## BindGlobal( "AL_ExponentsTrivialUnits", function( unit, one ) if unit = one then return [ 0 ]; else return [ 1 ]; fi; end ); BindGlobal( "ExponentsOfUnitsOfNumberField", function( F, elms ) local base, coef, exps, gens; # catch a trivial case if IsPrimeField(F) then # check whether all elements are units if ForAny( elms, x -> not x in [ One( F ), -One( F ) ] ) then return fail; fi; # return [ 0 ] for One( F ) and [ 1 ] for -One( F ) return List( elms, x -> AL_ExponentsTrivialUnits( x, One( F ))); fi; # determine exponents base := EquationOrderBasis( F ); coef := List( elms, x -> Coefficients( base, x ) ); exps := AL_FUNCS.ExponentsOfUnitsDescriptionWithRank( F, coef ); # add unit group if this is not yet known gens := List( exps.units, x -> LinearCombination( base, x ) ); AddUnitGroupOfNumberField( F, gens ); # return exponents if exps.expns = [ ] then return fail; else return exps.expns; fi; end ); InstallMethod( ExponentsOfUnits, "for number fields", true, [IsNumberField, IsCollection], 0, function( F, elms ) return ExponentsOfUnitsOfNumberField( F, elms ); end); ############################################################################# ## #F ExponentsOfUnitsWithRank( F, elms ) ## BindGlobal( "ExponentsOfUnitsWithRank", function( F, elms ) local base, flat, coef, exps, gens; # determine exponents base := EquationOrderBasis( F ); coef := List( elms, x -> Coefficients( base, x ) ); exps := AL_FUNCS.ExponentsOfUnitsDescriptionWithRank( F, coef ); # add unit group if this is not yet known gens := List( exps.units, x -> LinearCombination( base, x ) ); AddUnitGroupOfNumberField( F, gens ); # return exponents return rec(exps:=exps.expns, rank:=exps.rank); end ); ############################################################################# ## #F NormCosetsOfNumberField( F, norm ) ## InstallGlobalFunction( NormCosetsOfNumberField, function( F, norm ) local base, reps, gens; # catch a trivial case if IsPrimeField(F) then return [norm*One(F)]; fi; # get representatives mod unit group base := EquationOrderBasis( F ); reps := AL_FUNCS.NormCosetsDescription( F, norm ); # add unit group if this is not yet known gens := List( reps.units, x -> LinearCombination( base, x ) ); AddUnitGroupOfNumberField( F, gens ); # translate coset reps return List( reps.creps, x -> LinearCombination( base, x ) ); end ); ############################################################################# ## #M IsCyclotomicField( F ) ## InstallMethod( IsCyclotomicField, "for number fields", true, [IsNumberField], 0, function( F ) local U, t, o; U := UnitGroup(F); t := GeneratorsOfGroup(U)[1]; o := Order(t); return Phi(o) = DegreeOverPrimeField(F); end); alnuth-4.0.0/gap/isom.gi0000644000000000000000000000366215201210200011761 0ustar00############################################################################# ## #W isom.gi Alnuth - ALgebraic NUmber THeory Bjoern Assmann ## ############################################################################# ## #F IsomorphismOfMultGroupByFieldEl( F, elms ) ## BindGlobal( "IsomorphismOfMultGroupByFieldEl", function( F, elms ) local gens, rels, H, nat,CPCS,G; # calculate a constructive pc-sequence CPCS := CPCSOfGroupByFieldElements( F, elms ); H := PCPOfGroupByFieldElementsByCPCS( F, CPCS ); # new generating set for G := GroupByGenerators(CPCS.gens); nat := GroupHomomorphismByImagesNC( G, H, CPCS.gens, AsList(Pcp(H)) ); # add infos SetIsBijective( nat, true ); SetIsMultGroupByFieldElemsIsomorphism( nat, true ); nat!.CPCS := CPCS; nat!.field := F; return nat; end ); ############################################################################# ## #M Create isom to pcp group ## InstallOtherMethod( IsomorphismPcpGroup, "for matrix fields", true, [IsNumberFieldByMatrices, IsCollection], 0, function( F, elms ) return IsomorphismOfMultGroupByFieldEl( F, elms ); end); InstallOtherMethod( IsomorphismPcpGroup, "for fields def. by polynomial", true, [IsNumberField and IsAlgebraicExtension, IsCollection], 0, function( F, elms ) return IsomorphismOfMultGroupByFieldEl( F, elms ); end); ############################################################################# ## #M Images under group by field elems isom ## InstallMethod( ImagesRepresentative, "for group by field elems isom", FamSourceEqFamElm, [IsGroupGeneralMappingByImages and IsMultGroupByFieldElemsIsomorphism, IsMultiplicativeElementWithInverse], 0, function( nat, h ) local F, H, e, CPCS; F := nat!.field; CPCS := nat!.CPCS; H := Range( nat ); e := ExpVectorOfGroupByFieldElements( F, CPCS,h ); if e=fail then return fail; fi; return MappedVector( e, Pcp(H) ); end); alnuth-4.0.0/gap/matfield.gi0000644000000000000000000003330315201210200012572 0ustar00############################################################################# ## #W matfield.gi Alnuth - ALgebraic NUmber THeory Bettina Eick #W Bjoern Assmann #W Andreas Distler ## ############################################################################# ## #P IsNumberFieldByMatrices( ) ## InstallSubsetMaintenance( IsNumberFieldByMatrices, IsField and IsNumberFieldByMatrices, IsField ); ############################################################################# ## #F AL_SplitSemisimple( base ) ## BindGlobal( "AL_SplitSemisimple", function( base ) local d, b, f, s, i; d := Length( base ); b := PrimitiveAlgebraElement( [ ], base ); f := Factors( b.poly ); if Length( f ) = 1 then return [ rec( basis := IdentityMat( Length( b.elem ) ), poly := f ) ]; fi; s := List( f, function ( x ) return NullspaceRatMat( Value( x, b.elem ) ); end ); s := List( [ 1 .. Length( f ) ], function ( x ) return rec( basis := s[x], poly := f[x] ); end ); return s; end ); ############################################################################# ## #F AL_RadicalOfAbelianRMGroup( mats, d ) ## ## is an abelian rational matrix group ## BindGlobal( "AL_RadicalOfAbelianRMGroup", function( mats, d ) local coms, i, j, new, base, full, nath, indm, l, algb, newv, tmpb, subb, f, g, h, mat; base := []; full := IdentityMat( d ); # nath is the natural hom. from V to V/W nath := NaturalHomomorphismBySemiEchelonBases( full, base ); # indm for induced matrices indm := mats; # start spinning up basis and look for nilpotent elements i := 1; algb := []; while i <= Length( indm ) do # add next element to algebra basis l := Length( algb ); newv := Flat( indm[i] ); tmpb := SpinnUpEchelonBase(algb, [newv], indm{[1..i]},OnMatVector ); # check whether we have added a non-semi-simple element subb := []; for j in [l+1..Length(tmpb)] do mat := MatByVector( tmpb[j], Length(indm[i]) ); f := MinimalPolynomial( Rationals, mat ); g := Collected( Factors( f ) ); if ForAny( g, x -> x[2] > 1 ) then h := Product( List( g, x -> Value( x[1], mat ) ) ); Append( subb, List( h, x -> ShallowCopy(x) ) ); fi; od; #Print("found nilpotent submodule of dimension ", Length(subb),"\n"); # spinn up new subspace of radical subb := SpinnUpEchelonBase( [], subb, indm, OnRight ); if Length( subb ) > 0 then base := PreimageByNHSEB( subb, nath ); if Length( base ) = d then # radical cannot be so big return fail; fi; nath := NaturalHomomorphismBySemiEchelonBases( full, base ); indm := List( mats, x -> InducedActionFactorByNHSEB( x, nath ) ); algb := []; i := 1; else i := i + 1; fi; od; return rec( radical := base, nathom := nath, algebra := algb ); end ); ############################################################################# ## #F AL_HomogeneousSeriesAbelianRMGroup( mats, d ) ## ## is an abelian rational matrix group ## DeclareGlobalFunction( "AL_HomogeneousSeriesAbelianRMGroup" ); InstallGlobalFunction( "AL_HomogeneousSeriesAbelianRMGroup", function( mats, d ) local radb, splt, nath,inducedgens, l, sers, i, sub, full, acts, rads; # catch the trivial case and set up if d = 0 then return []; fi; full := IdentityMat( d ); if Length( mats ) = 0 then return [full, []]; fi; sers := [full]; # get the radical radb := AL_RadicalOfAbelianRMGroup( mats, d ); if radb = fail then return fail; fi; splt := AL_SplitSemisimple( radb.algebra ); nath := radb.nathom; # refine radical factor and initialize series l := Length( splt ); for i in [2..l] do sub := Concatenation( List( [i..l], x -> splt[x].basis ) ); TriangulizeMat( sub ); Add( sers, PreimageByNHSEB( sub, nath ) ); od; Add( sers, radb.radical ); # induce action to radical nath := NaturalHomomorphismBySemiEchelonBases( full, radb.radical); acts := List( mats, x -> InducedActionSubspaceByNHSEB( x, nath )); # use recursive call to refine radical rads := AL_HomogeneousSeriesAbelianRMGroup( acts, Length(radb.radical) ); if rads = fail then return fail; fi; rads := List( rads, function(x) if x=[] then return []; else return x * radb.radical; fi;end ); Append( sers, rads{[2..Length(rads)]} ); return sers; end ); ############################################################################# ## #F AL_MatricesGeneratingNumberField( gens ) ## BindGlobal( "AL_MatricesGeneratingNumberField", function( gens ) local d, series, G; d := Length(gens[1]); if ForAny( gens, x -> Length(x) <> d ) then Print("matrices must have same dimensions\n"); return false; elif not ForAll( Flat( gens ), IsRat ) then Print("matrices must be rational\n"); return false; elif ForAny( gens, x -> RankMat(x) <> d ) then Print("matrices must be invertible \n"); return false; fi; G := Group( gens ); if not IsAbelian( G ) then Print( "The algebra generated by the matrices is not abelian\n" ); return false; fi; series := AL_HomogeneousSeriesAbelianRMGroup( gens, d ); if Length( series ) > 2 then Print( "Matrices do not generate a field.\n" ); Print( "The natural module Q^",d," is not homogeneous.\n" ); return false; fi; return true; end ); ############################################################################# ## #F FieldByMatrices( gens ) ## InstallGlobalFunction( FieldByMatricesNC, function( gens ) local F; F := FieldByGenerators( gens); SetIsNumberField( F, true ); SetIsNumberFieldByMatrices( F, true ); SetIsFinite( F, false ); return F; end ); InstallGlobalFunction( FieldByMatrices, function( gens ) local F; if not AL_MatricesGeneratingNumberField( gens ) then return fail; fi; F := FieldByMatricesNC( gens ); DegreeOverPrimeField( F ); return F; end ); ############################################################################# ## #F FieldByMatrixBasisNC( gens ) . . . MatFieldByAlgebraBasis ## InstallGlobalFunction( FieldByMatrixBasisNC, function( gens ) local F, B; F := FieldByMatricesNC( gens ); B := Objectify(NewType(FamilyObj(F), IsBasisOfMatrixField), rec()); SetUnderlyingLeftModule( B, F ); SetBasisVectors( B, gens ); SetBasis( F, B ); DegreeOverPrimeField( F ); return F; end ); InstallGlobalFunction( FieldByMatrixBasis, function( gens ) local V; if not AL_MatricesGeneratingNumberField( gens ) then return fail; fi; V := VectorSpace( Rationals, gens ); # test linear independence if Length( Basis( V )) < Length( gens ) then Print("matrices must be linearly independent\n"); return fail; fi; # test whether V = Field( gens ) if Length( AlgebraBase( gens )) > Length( gens ) then Print("dimension of generated field is greater than vector space dimension\n"); return fail; fi; return FieldByMatrixBasisNC( gens ); end ); ############################################################################# ## #F BasisVectorsOfMatrixField( F ) #M CanonicalBasis( F ) #M Basis( F ) ## BindGlobal( "BasisVectorsOfMatrixField", function( F ) return AlgebraBase( GeneratorsOfField(F) ); end ); InstallMethod( CanonicalBasis, "for matrix field", true, [IsNumberFieldByMatrices], 0, function( F ) local B, b; B := Objectify(NewType(FamilyObj(F), IsBasisOfMatrixField), rec()); b := BasisVectorsOfMatrixField( F ); SetUnderlyingLeftModule( B, F ); SetBasisVectors( B, b ); return B; end ); InstallMethod( Basis, "for matrix field", true, [IsNumberFieldByMatrices], 0, function( F ) return CanonicalBasis( F ); end ); ############################################################################# ## #M Coefficients( B, a ) ## InstallMethod( Coefficients, "for basis of matrix field", true, [IsBasisOfMatrixField, IsVector ], 15, function( B, a ) local b; b := BasisVectors( B ); b := List( b, Flat ); return SolutionMat( b, Flat(a) ); end ); ############################################################################# ## #M DegreeOverPrimeField( F ) ## InstallMethod( DegreeOverPrimeField, "for matrix field", true, [IsNumberFieldByMatrices], 0, function( F ) return Length( Basis( F ) ); end); ############################################################################# ## #F IntegralMatrix #F SuitablePrimitiveElementCheck( F, k ) ## BindGlobal( "IntegralMatrix", function( mat ) local l,n,i,j,a; l := []; n := Length( mat ); for i in [1..n] do for j in [1..n] do Add( l, DenominatorRat( mat[i][j]) ); od; od; a := Lcm( l ); return a*mat; end ); BindGlobal( "SuitablePrimitiveElementCheck", function( F, k ) local d, g, sumCoef; d := DegreeOverPrimeField( F ); g := MinimalPolynomial( Rationals, k ); if not Degree(g) = d then return false; elif ForAll( CoefficientsOfUnivariatePolynomial(g), IsInt ) then sumCoef := Sum( List(CoefficientsOfUnivariatePolynomial(g),x->AbsInt(x)) ); Info( InfoAlnuth, 3, "sum of the coefficients is"); Info( InfoAlnuth, 3, sumCoef ); return rec( prim := k, min := g, sumCoef := sumCoef); else k := IntegralMatrix( k ); g := MinimalPolynomial( Rationals, k ); sumCoef := Sum( List(CoefficientsOfUnivariatePolynomial(g),x->AbsInt(x)) ); Info( InfoAlnuth, 3, "sum of the coefficients is"); Info( InfoAlnuth, 3, sumCoef ); return rec( prim := k, min := g, sumCoef := sumCoef); fi; end ); ############################################################################# ## #F SuitablePrimitiveElementOfMatrixField( F ) #M IntegerPrimitiveElement( F ) #M PrimitiveElement( F ) ## BindGlobal( "SuitablePrimitiveElementOfMatrixField", function( F ) local k, d, b, l, i, c, primtmp,prim, poss; # try to find a primitive element wiht small coeff in the minpol prim := rec( prim := [], min := [], sumCoef := infinity); poss := 0; #catch the trivial case if DegreeOverPrimeField(F)=1 then return One(F); fi; # first try the generators of F for k in GeneratorsOfField(F) do primtmp := SuitablePrimitiveElementCheck( F, k ); if not IsBool( primtmp ) then poss := poss + 1; if primtmp.sumCoef < prim.sumCoef then prim := primtmp; fi; fi; od; # otherwise try random elements d := DegreeOverPrimeField( F ); b := Basis(F); l := List( [1..d], x -> 0 ); Append( l, [1,1,-1] ); i := 1; while poss < AL_CurrentPrimitiveElementTrials() do Info( InfoAlnuth, 3, "another try to calculate primitive element"); Info( InfoAlnuth, 3, i ); c := List( [1..d], x -> RandomList( l ) ); k := LinearCombination( b, c ); primtmp := SuitablePrimitiveElementCheck( F, k ); if not IsBool( primtmp ) then poss := poss + 1; if primtmp.sumCoef < prim.sumCoef then prim := primtmp; fi; fi; i := i + 1; Append( l, [i,i,-i] ); od; SetDefiningPolynomial( F, prim.min ); SetIntegerDefiningPolynomial( F, prim.min ); Info( InfoAlnuth, 2, "prim is ", prim); return prim.prim; end ); InstallMethod( IntegerPrimitiveElement, "for matrix field", true, [IsNumberFieldByMatrices], 0, function( F ) return SuitablePrimitiveElementOfMatrixField(F); end); InstallMethod( PrimitiveElement, "for matrix field", true, [IsNumberFieldByMatrices], 0, function( F ) return IntegerPrimitiveElement(F); end); InstallOtherMethod( DefiningPolynomial, "for matrix field", true, [IsNumberFieldByMatrices], 0, function( F ) return MinimalPolynomial( Rationals, PrimitiveElement(F) ); end); ############################################################################# ## #M IntegerDefiningPolynomial( F ) ## InstallMethod( IntegerDefiningPolynomial, "for matrix field", true, [IsNumberFieldByMatrices], 0, function( F ) return MinimalPolynomial( Rationals, IntegerPrimitiveElement( F ) ); end ); ############################################################################# ## #F Norm(F,k) ## InstallOtherMethod( Norm, "for matrix fields", true, [IsNumberFieldByMatrices, IsMultiplicativeElement], SUM_FLAGS, function( F, k ) local l, d; l := Length(k); d := DegreeOverPrimeField(F); return Root( Determinant(k), l/d ); end ); ############################################################################# ## #M PrintObj( F ) #M ViewObj( F ) ## InstallMethod( PrintObj, "for a matrix field", true, [IsNumberFieldByMatrices], 0, function( F ) if HasDegreeOverPrimeField( F ) then Print( "" ); else Print(""); fi; end ); InstallMethod( ViewObj, "for a matrix field", true, [IsNumberFieldByMatrices], 0, function( F ) if HasDegreeOverPrimeField( F ) then Print( "" ); else Print(""); fi; end ); alnuth-4.0.0/gap/matunits.gi0000644000000000000000000000522215201210200012650 0ustar00############################################################################# ## #W matunits.gi Alnuth - ALgebraic NUmber THeory Bettina Eick ## ############################################################################# ## #F IntersectionOfUnitSubgroups( F, gen1, gen2 ) ## ## Let and be two subgroups of U(F). ## This function computes the intersection of and as exponents ## in the generators of . ## InstallGlobalFunction( IntersectionOfUnitSubgroups, function(F,gen1,gen2) local add, ad1, ad2, U, u, l, m, t, int, i; # get an additive description of both unit groups add := ExponentsOfUnits( F, Concatenation( gen1, gen2 ) ); ad1 := add{[1..Length(gen1)]}; ad2 := add{[Length(gen1)+1..Length(gen1)+Length(gen2)]}; # extract units U := UnitGroupOfNumberField( F ); u := GeneratorsOfGroup(U); l := Length(u)-1; m := Order(u[1]); t := List( [1..l+1], x -> 0 ); t[1] := m; # compute intersection int := NullspaceIntMat( Concatenation( ad1, ad2, [t] ) ); int := int{[1..Length(int)]}{[1..Length(ad1)]}; # reduce torsion if int[1][1] <> 0 then int[1][1] := int[1][1] mod m; fi; if int[1] = 0*int[1] then int := int{[2..Length(int)]}; fi; return int; end ); ############################################################################# ## #F IntersectionOfTFUnitsByCosets( F, mats, C ) ## ## is a torsion free subgroup of U(F) und C are some cosets in U(F). ## InstallGlobalFunction( IntersectionOfTFUnitsByCosets, function( F, mats, C ) local a, b, c, d, all, add, rep, s, i, r, fr1, fr2, int; # set up a := Length( mats ); b := Length( C.units ); c := Length( C.reprs ); # determine additive description all := Concatenation( mats, C.units, C.reprs ); add := ExponentsOfUnits( F, all ); # mod out torsion d := Length( GeneratorsOfGroup(UnitGroup(F)) ) - 1; add := add{[1..Length(add)]}{[2..d+1]}; # cut into pieces rep := add{[ a+b+1..a+b+c ]}; add := add{[ 1..a+b ]}; # loop over reps s := false; i := 1; while IsBool(s) and i <= Length(rep) do r := SolutionIntMat( add, rep[i] ); i := i+1; if not IsBool(r) then s := r{[1..a]}; fi; od; # if we cannot find a representative, then return if IsBool(s) then return false; fi; # otherwise add intersection of mats with C.units fr1 := add{[1..a]}; fr2 := add{[a+1..a+b]}; # find linear combinations of mgrp in unit int := NullspaceIntMat( Concatenation( fr1, fr2 ) ); int := int{[1..Length(int)]}{[1..a]}; return rec( repr := s, ints := int ); end ); alnuth-4.0.0/gap/oscar.gi0000644000000000000000000001047015201210200012114 0ustar00############################################################################# ## #W oscar.gi Alnuth - ALgebraic NUmber THeory Max Horn #W Claus Fieker ## ## ## This file is a drop-in replacement for pari.gi, thus alls functions are ## also made available with "Pari" in their name. BindGlobal("OSCAR_AL_FUNCS", rec()); # store isomorphism between GAP and OSCAR univariate polynomial ring over the # rationals as we need it again and again, for each new number field OSCAR_AL_FUNCS.PolyRingIso := Oscar.iso_gap_oscar(PolynomialRing(Rationals)); # Given a GAP number field, return an isomorphic Oscar number field. # This field is also cached OSCAR_AL_FUNCS.OscarField := function(F) local f; if not IsBound(F!.oscarField) then f := OSCAR_AL_FUNCS.PolyRingIso(IntegerDefiningPolynomial(F)); F!.oscarField := Oscar.number_field(f)[1]; fi; return F!.oscarField; end; OSCAR_AL_FUNCS.MaximalOrderDescription := function(F) local K, O, basis; K := OSCAR_AL_FUNCS.OscarField(F); O := Oscar.maximal_order(K); basis := Julia.map(Oscar.coordinates,Oscar.basis(O,K)); return JuliaToGAP(IsList, basis, true); end; OSCAR_AL_FUNCS.UnitGroupDescription := function(F) local K, O, U_m, U, m, basis; K := OSCAR_AL_FUNCS.OscarField(F); O := Oscar.maximal_order(K); U_m := Oscar.unit_group(O); U := U_m[1]; m := U_m[2]; basis := Julia.map(Oscar.coordinates, Julia.map(K, Julia.map(m, Oscar.gens(U)))); return JuliaToGAP(IsList, basis, true); end; OSCAR_AL_FUNCS.ExponentsOfUnitsDescriptionWithRank := function(F, elms) local K, O, U_m, U, m, basis, units, rank, expns, x; K := OSCAR_AL_FUNCS.OscarField(F); O := Oscar.maximal_order(K); U_m := Oscar.unit_group(O); U := U_m[1]; m := U_m[2]; basis := Julia.map(Oscar.coordinates, Julia.map(K, Julia.map(m, Oscar.gens(U)))); units := JuliaToGAP(IsList, basis, true); # the order of the torsion part of the full unit group rank := Oscar.GAP.julia_to_gap(Oscar.order(U_m[1][1])); expns := []; for x in elms do # map GAP int vector to Oscar element Add(expns, Oscar.GAP.julia_to_gap(Oscar.preimage(m, K(GAPToJulia(x))).coeff)[1]); od; # return result return rec(units := units, expns := expns, rank := rank); end; OSCAR_AL_FUNCS.ExponentsOfFractionalIdealDescription := function(F, elms) local K, O, tmp, ideals, result, blah, vec, I; K := OSCAR_AL_FUNCS.OscarField(F); O := Oscar.maximal_order(K); tmp := Julia.map(x -> Oscar.factor(K(GAPToJulia(x)) * O), elms); # take the union of the keys ideals := Julia.collect(Julia.reduce(Oscar.union, Julia.map(Oscar.Set, Julia.map(Oscar.keys, tmp)))); if Julia.isempty(ideals) then return []; fi; result := []; for blah in JuliaToGAP(IsList, tmp) do vec := []; for I in JuliaToGAP(IsList, ideals) do Add(vec, Julia.get(blah, I, 0)); od; Add(result, vec); od; return result; end; OSCAR_AL_FUNCS.NormCosetsDescription := function(F, norm) local K, O, U_m, U, m, basis, units, eqn, creps; K := OSCAR_AL_FUNCS.OscarField(F); O := Oscar.maximal_order(K); U_m := Oscar.unit_group(O); U := U_m[1]; m := U_m[2]; basis := Julia.map(Oscar.coordinates, Julia.map(K, Julia.map(m, Oscar.gens(U)))); units := JuliaToGAP(IsList, basis, true); eqn := Oscar.norm_equation(O, norm); creps := JuliaToGAP(IsList, Julia.map(Oscar.coordinates, Julia.map(K, eqn)), true); # return result return rec(units := units, creps := creps); end; OSCAR_AL_FUNCS.Vector_nf_elem := JuliaType( Julia.Vector, [ Oscar_jl.nf_elem ] ); OSCAR_AL_FUNCS.PolynomialFactorsDescription := function(F, coeffs) local K, cf, poly, facs, result, f, g, i; K := OSCAR_AL_FUNCS.OscarField(F); cf := OSCAR_AL_FUNCS.Vector_nf_elem(Reversed(List(coeffs, x -> K(GAPToJulia(x))))); poly := Oscar.polynomial(K, cf); facs := Oscar.factor(poly); Assert(0, Oscar.is_one(facs.unit)); result := []; for f in JuliaToGAP(IsList, Oscar.collect(facs.fac)) do # Convert factor to GAP g := Julia.map(Oscar.coordinates, f[1].coeffs); g := JuliaToGAP(IsList, g, true); g := Reversed(g); # add as many copies as necessary for i in [1..f[2]] do Add(result, g); od; od; # Sort result by ascending degrees SortBy(result, Length); return result; end; AL_FUNCS := OSCAR_AL_FUNCS; alnuth-4.0.0/gap/pari.gi0000644000000000000000000001650615201210200011746 0ustar00############################################################################# ## #W pari.gi Alnuth - ALgebraic NUmber THeory Bettina Eick #W Bjoern Assmann #W Andreas Distler ## ############################################################################# ## #F PolynomialWithNameToStringList( f[, name] ) ## BindGlobal("PolynomialWithNameToStringList", function( arg ) local c, f, stringlist, i; # get input f := arg[1]; # print identifier of polynomial, default 'f', or given as second argument if Length( arg ) = 1 then stringlist := ["f = "]; else stringlist := Concatenation(arg[2], " = "); fi; # print polynomial using 'x' as variable c := CoefficientsOfUnivariatePolynomial( f ); for i in [1..Length(c)] do if c[i] >= 0 and i > 1 then Add(stringlist, "+"); fi; Add(stringlist, Concatenation(String(c[i]), "*x^", String(i-1))); od; Add(stringlist,";\n"); return stringlist; end); ############################################################################# ## #F CoefficientsToStringList(name, coeffs) ## BindGlobal("CoefficientsToStringList", function(name, coeffs) local stringlist, c; stringlist := ["{"]; Add(stringlist, name); Add(stringlist, "= [ \n"); for c in coeffs do Add(stringlist, String(c)); Add(stringlist, ", \n"); od; Add(stringlist, "0]; }\n"); return stringlist; end); ############################################################################# ## #F ProcessPariGP(input, codefile) ## BindGlobal("ProcessPariGP", function(input, codefile) local exe, output, paricode, memopt; exe := AL_CurrentPariGpPath(); if exe = fail then Error("Alnuth couldn't locate a usable PARI/GP executable.\n", "You may the `PariGpPath` user preference to point to one."); fi; # add the prepared code fragments for the calculations in PARI/GP paricode := InputTextFile( Filename(DirectoriesPackageLibrary("alnuth", "gp"), codefile) ); # execute PARI/GP Info(InfoAlnuth, 1, "executing PARI/GP with ", codefile); output := ""; memopt := Concatenation("-s", String(AL_CurrentPariStackSize()), "M"); Process(DirectoryCurrent(), exe, InputTextString(Concatenation(input, ReadAll(paricode))), OutputTextString(output,false), Concatenation(AL_CurrentPariGpOptions(), [memopt]) ); # close open input stream from file with GP code CloseStream(paricode); return EvalString(output); end); ############################################################################# ## #F MaximalOrderDescriptionPari( F ) ## BindGlobal("MaximalOrderDescriptionPari", function( F ) local input, result; if IsPrimeField(F) then return [1]; fi; # initialize list of input strings with the defining polynomial input := PolynomialWithNameToStringList(IntegerDefiningPolynomial(F)); # execute PARI/GP result := ProcessPariGP(Concatenation(input), "maxord.gp"); # return result return result; end); ############################################################################# ## #F UnitGroupDescriptionPari( F ) ## BindGlobal("UnitGroupDescriptionPari", function( F ) local input, result; if IsPrimeField( F ) then return [-1]; fi; # initialize list of input strings with the defining polynomial input := PolynomialWithNameToStringList(IntegerDefiningPolynomial(F)); # execute PARI/GP result := ProcessPariGP(Concatenation(input), "units.gp"); # return result return result; end); ############################################################################# ## #F ExponentsOfUnitsDescriptionWithRankPari( F, elms ) ## BindGlobal("ExponentsOfUnitsDescriptionWithRankPari", function( F, elms ) local input, e, result; if IsPrimeField( F ) then return fail; fi; # initialize list of input strings with the defining polynomial input := PolynomialWithNameToStringList(IntegerDefiningPolynomial(F)); # add coefficients of input := Concatenation(input, CoefficientsToStringList("elms", elms)); # execute PARI/GP result := ProcessPariGP(Concatenation(input), "decompra.gp"); # return result return rec(units := result[1], expns := result[2], rank := result[3]); end); ############################################################################# ## #F ExponentsOfFractionalIdealDescriptionPari( F, elms ) ## ## are arbitrary elements of F (or rather their coefficients). ## Returns the exponents vectors of the fractional ideals ## generated by elms corresponding to the underlying prime ideals. ## BindGlobal( "ExponentsOfFractionalIdealDescriptionPari", function( F, elms ) local input, e, result; if IsPrimeField( F ) then return fail; fi; # initialize list of input strings with the defining polynomial input := PolynomialWithNameToStringList(IntegerDefiningPolynomial(F)); # add coefficients of input := Concatenation(input, CoefficientsToStringList("elms", elms)); # execute PARI/GP result := ProcessPariGP(Concatenation(input), "fracidea.gp"); # return result return result; end); ############################################################################# ## #F NormCosetsDescriptionPari( F, norm ) ## BindGlobal( "NormCosetsDescriptionPari", function( F, norm ) local input, result; if IsPrimeField(F) then return fail; fi; # initialize list of input strings with the defining polynomial input := PolynomialWithNameToStringList(IntegerDefiningPolynomial(F)); # add norm information Add(input, "nrm = "); Add(input, String(norm)); Add(input, "; \n"); # execute PARI/GP result := ProcessPariGP(Concatenation(input), "norm.gp"); # return result return rec(units := result[1], creps := result[2]); end); ############################################################################# ## #F PolynomialFactorsDescriptionPari, function( , ) ## ## Factorizes the polynomial defined by over the field ## using PARI/GP ## BindGlobal( "PolynomialFactorsDescriptionPari", function( F, coeffs ) local input, c, result, runtime; # initialize list of input strings with the defining polynomial input := PolynomialWithNameToStringList(IntegerDefiningPolynomial(F)); # add the coefficients of the polynomial to be factorised input := Concatenation(input, CoefficientsToStringList("coeffs", coeffs)); # execute PARI/GP result := ProcessPariGP(Concatenation(input), "polyfactors.gp"); # print runtime runtime := Remove(result); Info(InfoAlnuth, 1, "Runtime: ", runtime); # return result return result; end ); BindGlobal("PARI_AL_FUNCS", rec( MaximalOrderDescription := MaximalOrderDescriptionPari, UnitGroupDescription := UnitGroupDescriptionPari, ExponentsOfUnitsDescriptionWithRank := ExponentsOfUnitsDescriptionWithRankPari, ExponentsOfFractionalIdealDescription := ExponentsOfFractionalIdealDescriptionPari, NormCosetsDescription := NormCosetsDescriptionPari, PolynomialFactorsDescription := PolynomialFactorsDescriptionPari, )); AL_FUNCS := PARI_AL_FUNCS; alnuth-4.0.0/gap/polfield.gi0000644000000000000000000000434715201210200012611 0ustar00############################################################################# ## #W polfield.gi Alnuth - ALgebraic NUmber THeory Bettina Eick #W Bjoern Assmann ## ############################################################################# ## #F FieldOfPolynomial(f) ## InstallGlobalFunction( FieldOfPolynomial, function( f ) local c; c := CoefficientsOfUnivariatePolynomial(f); return Field(c); end ); ############################################################################# ## #F FieldByPolynomial( f ) ## InstallGlobalFunction( FieldByPolynomialNC, function( f ) return AlgebraicExtension( Rationals, f ); end ); InstallGlobalFunction( FieldByPolynomial, function( f ) if DegreeOfUnivariateLaurentPolynomial(f) <= 0 then Print("polynomial must have degree at least 1\n"); return fail; fi; if not IsIrreducible( f ) then Print("polynomial must be irreducible\n"); return fail; fi; if not ForAll( CoefficientsOfUnivariatePolynomial( f ), IsRat ) then Print("polynomial must be defined over Q \n"); return fail; fi; return FieldByPolynomialNC(f); end ); ############################################################################# ## #M IntegerPrimitiveElement( F ) ## InstallMethod( IntegerPrimitiveElement, "for algebraic extension", true, [IsNumberField and IsAlgebraicExtension], 0, function( F ) local coeff; coeff := CoefficientsOfUnivariatePolynomial( DefiningPolynomial( F )); # AD improvement possible, e.g. x^5 - 1/32 return Lcm( List( coeff, DenominatorRat ) ) * PrimitiveElement( F ); end ); ############################################################################# ## #M IntegerDefiningPolynomial( F ) ## InstallMethod( IntegerDefiningPolynomial, "for algebraic extension", true, [IsNumberField and IsAlgebraicExtension], 0, function( F ) local f, c, k, n; c := CoefficientsOfUnivariatePolynomial(DefiningPolynomial(F)); k := ExtRepOfObj( IntegerPrimitiveElement(F)/PrimitiveElement(F) )[1]; n := Degree( DefiningPolynomial(F) ); c := List( [0..n], i -> c[i+1] * k^(n-i) ); return UnivariatePolynomial( Rationals, c ); end ); alnuth-4.0.0/gap/present.gi0000644000000000000000000002177615201210200012500 0ustar00############################################################################# ## #W present.gi Alnuth - ALgebraic NUmber THeory Bjoern Assmann ## ############################################################################# ## #F GeneratorLatticeTF( rels ) ## ## rels is a sublattice of Z^n. ## this function compute a minimal generatorset of Z^n/rels ## which is torsion-free ## BindGlobal( "GeneratorLatticeTF", function( rels ) local n, normal,i,gens; n := Length( rels[1] ); normal:=NormalFormIntMat(rels,13); for i in [1..Length( normal.normal )] do if normal.normal[i][i] <> 1 then break; fi; od; if i = Length( normal.normal ) then i := i+1; fi; #gens := IdentityMat( n ); #gens := gens{[i..n]}; #gens := gens * normal.coltrans^-1; gens := (normal.coltrans^-1); gens := gens{[i..n]}; return gens; end ); ############################################################################# ## #F CPCSOfTFGroupByFieldElements( F, gens ) ## ## computes a constructive polycyclic sequence for gens ## BindGlobal( "CPCSOfTFGroupByFieldElements", function( F, gens ) local rels, exps, newGens, e, newRels; rels := RelationLatticeTF( F, gens ); if rels = [] then return rec( gens:=gens, relord:=List([1..Length(gens)], x -> 0)); fi; exps := GeneratorLatticeTF(rels); newGens := []; for e in exps do Add( newGens, MappedVector( e, gens )); od; newRels := List( [1..Length(newGens)], x -> 0); return rec( gens:=newGens, relord:=newRels); end ); ############################################################################# ## #F ExpVectorOfTFGroupByFieldElements( F, CPCS, elm ) ## ## torsion free group !! ## BindGlobal( "ExpVectorOfTFGroupByFieldElements", function( F, CPCS, elm ) local elms, rels, v, m, exp, i; m := Length( CPCS.gens ); elms := StructuralCopy( CPCS.gens ); Add( elms, elm ); rels:= RelationLatticeTF( F, elms ); v := rels[1]; exp:=[]; for i in [1..m] do exp[i]:=-v[i]/v[m+1]; od; Assert( 2, MappedVector( exp, CPCS.gens ) = elm ); return exp; end ); ############################################################################# ## #F Exp2genList(exp); ## BindGlobal( "Exp2GenList", function(exp) local n, genList,i; n:=Length(exp); genList:=[]; for i in [1..n] do if exp[i] <> 0 then Append(genList,[i,exp[i]]); fi; od; return genList; end ); ############################################################################# ## #F PCPOfTFGroupByFieldElementsByCPCS( F, CPCS ) ## BindGlobal( "PCPOfTFGroupByFieldElementsByCPCS", function( F, CPCS ) local ro, n; # Setup ro := CPCS.relord; n := Length( CPCS.gens ); return AbelianPcpGroup( n, ro ); end ); ############################################################################# ## #F PcpPresentationOfTFGroupByFieldElements( F, elms ) ## ## given a field F and elms in F. ## this function computes a polycyclic presentation for the group ## generated by elms ## BindGlobal( "PcpPresentationOfTFGroupByFieldElements", function( F, elms ) local CPCS, pcp; CPCS := CPCSOfTFGroupByFieldElements( F, elms ); pcp := PCPOfTFGroupByFieldElementsByCPCS( F, CPCS ); return pcp; end ); ############################################################################# # generalization to groups with torsion ############################################################################# ## #F GeneratorLattice( rels ) ## ## rels is a sublattice of Z^n. ## this function compute a minimal generatorset of Z^n/rels ## and the orders of the generators. ## BindGlobal( "GeneratorLattice", function( rels ) local n, normal,i,gens, torsion, relord, j; torsion := false; n := Length( rels[1] ); normal:=NormalFormIntMat(rels,13); for i in [1..Length( normal.normal )] do if normal.normal[i][i] <> 1 then torsion := true; break; fi; od; if not torsion then i := i+1; fi; #gens := IdentityMat( n ); #gens := gens{[i..n]}; #gens := gens * normal.coltrans^-1; gens := (normal.coltrans^-1); gens := gens{[i..n]}; # set relative orders of gens relord := []; for j in [i..n] do if IsBound(normal.normal[j]) then Add( relord, normal.normal[j][j] ); else Add( relord, 0 ); fi; od; return rec(exps := gens, relord:=relord ); end ); ############################################################################# ## #F CPCSOfGroupByFieldElements( F, gens ) ## ## computes a constructive polycyclic sequence for gens ## BindGlobal( "CPCSOfGroupByFieldElements", function( F, gens ) local rels, exps, newGens, e, newRels; rels := RelationLattice( F, gens ); if rels = [] then return rec( gens:=gens, relord:=List([1..Length(gens)], x -> 0)); fi; exps := GeneratorLattice(rels); newGens := []; for e in exps.exps do Add( newGens, MappedVector( e, gens )); od; return rec( gens:=newGens, relord:=exps.relord); end ); ############################################################################# ## #F ExpVectorOfGroupByFieldElements( F, CPCS, elm ) ## BindGlobal( "ExpVectorOfGroupByFieldElements", function( F, CPCS, elm ) local elms, rels, l, m, exp, i; if IsList(CPCS.gens[1]) then if Length(elm) <> Length(CPCS.gens[1]) then return fail; fi; fi; m := Length( CPCS.gens ); if elm=elm^0 then return List([1..m], x-> 0); fi; elms := StructuralCopy( CPCS.gens ); l :=Concatenation([elm],elms); rels := RelationLattice( F, l ); if rels=[] then return fail; fi; rels := NormalFormIntMat(rels,0).normal; if not rels[1][1]=1 then return fail; fi; exp := -rels[1]{[2..(m+1)]}; Assert( 2, MappedVector( exp, CPCS.gens ) = elm, "failure in ExpVectorOfGroupByFieldElements" ); return exp; end ); ############################################################################# ## #F PCPOfGroupByFieldElementsByCPCS( F, CPCS ) ## BindGlobal( "PCPOfGroupByFieldElementsByCPCS", function( F, CPCS ) local ro, n; # Setup ro := CPCS.relord; n := Length( CPCS.gens ); return AbelianPcpGroup( n, ro ); end ); ############################################################################# ## #F PcpPresentationOfGroupByFieldElements( F, elms ) ## ## given a field F and elms in F. ## this function computes a polycyclic presentation for the group ## generated by elms ## BindGlobal( "PcpPresentationOfGroupByFieldElements", function( F, elms ) local CPCS, pcp; CPCS := CPCSOfGroupByFieldElements( F, elms ); pcp := PCPOfGroupByFieldElementsByCPCS( F, CPCS ); return pcp; end ); InstallGlobalFunction( PcpPresentationMultGroupByFieldEl, function( F, elms ) local CPCS, pcp; CPCS := CPCSOfGroupByFieldElements( F, elms ); pcp := PCPOfGroupByFieldElementsByCPCS( F, CPCS ); return pcp; end); InstallGlobalFunction( PcpPresentationOfMultiplicativeSubgroup, function( F, elms ) local CPCS, pcp; CPCS := CPCSOfGroupByFieldElements( F, elms ); pcp := PCPOfGroupByFieldElementsByCPCS( F, CPCS ); return pcp; end); ############################################################################# # test functions BindGlobal( "RandomGroupElement", function(gens) local d,k,g,i,length,x,n; k:=Length(gens); g:=gens[1]^0; length:=Random(5,30); for i in [1..length] do x:=Random(1,k); n:=Random(Integers); g:=g*(gens[x]^n); od; return g; end ); BindGlobal( "TestExpVectorOfGroupByFieldElements", function( F, CPCS, gens,numberOfTests) local i,g,exp; SetAssertionLevel(2); for i in [1..numberOfTests] do Print(i); g:=RandomGroupElement(gens); #Print("g ist gleich ",g,"\n"); exp:=ExpVectorOfGroupByFieldElements( F, CPCS, g ); #Print("exp ist gleich ",exp,"\n"); od; SetAssertionLevel(0); end ); BindGlobal( "TestCPCSOfGroupByFieldElements", function( mats ) local F,j, CPCS; F := FieldByMatricesNC( mats ); #j := Random([1..Length(mats)]); j := Length(mats); CPCS:=CPCSOfGroupByFieldElements( F, mats{[1..j]} ); TestExpVectorOfGroupByFieldElements( F, CPCS, mats{[1..j]},10); end ); # faster BindGlobal( "TestCPCSOfGroupByFieldElements2", function( mats ) local F,j, CPCS,b; F := FieldByMatricesNC( mats ); b := Basis( F ); F := FieldByMatrixBasisNC( b ); #j := Random([1..Length(mats)]); j := Length(mats); CPCS:=CPCSOfGroupByFieldElements( F, mats{[1..j]} ); TestExpVectorOfGroupByFieldElements( F, CPCS, mats{[1..j]},10); end ); BindGlobal( "TestCPCSOfGroupByFieldElementsPol", function( pol ) local elms,F,j, CPCS; F := FieldByPolynomialNC( pol ); elms := List( [1..10], x-> Random(F) ); CPCS:=CPCSOfGroupByFieldElements( F, elms ); TestExpVectorOfGroupByFieldElements( F, CPCS, elms,10); end ); alnuth-4.0.0/gap/rationals.gi0000644000000000000000000000133315201210200012777 0ustar00############################################################################# ## #W rationals.gi Alnuth - ALgebraic NUmber THeory Andreas Distler ## ############################################################################# ## #M IntegerPrimitiveElement( Rationals ) ## InstallMethod( IntegerPrimitiveElement, "for the rationals", true, [IsRationals], 0, One ); ############################################################################# ## #M EquationOrderBasis( Rationals ) #M MaximalOrderBasis( Rationals ) ## InstallMethod( EquationOrderBasis, "for the rationals", true, [IsRationals], 15, CanonicalBasis ); InstallMethod( MaximalOrderBasis, "for the rationals", true, [IsRationals], 15, CanonicalBasis ); alnuth-4.0.0/gap/rels.gi0000644000000000000000000001664615201210200011765 0ustar00############################################################################# ## #W rels.gi Alnuth - ALgebraic NUmber THeory Bjoern Assmann ## ############################################################################# ## #F RelationLatticeOfTFUnits( F, elms ) ## ## The input is a list of elements which generate a free abelian subgroup ## of the unit group of F. The function computes the relation lattice of ## . ## InstallGlobalFunction( RelationLatticeOfTFUnits, function( F, elms ) local exps,rels,l; # do a simple check if ForAll( elms, x -> x = x^0 ) then return IdentityMat( Length(elms) ); fi; # now compute an additive description exps := ExponentsOfUnits( F, elms ); Info(InfoAlnuth,2,exps); Info(InfoAlnuth,2,"exps"); # the first entry in the vectors of exps corresponds to torsion - # mod out l := Length( exps[1] ); exps := exps{[1..Length(exps)]}{[2..l]}; # compute the relations rels := NullspaceIntMat( exps ); # format results if Length(rels) = 0 then return rels; fi; return NormalFormIntMat( rels, 2 ).normal; end ); ############################################################################# ## #F NullspaceModRank( M, n) ## BindGlobal( "NullspaceModRank", function( M, n ) local snf, null, nullM, i, gcdex; snf := NormalFormIntMat( M, 1 + 4 ); null := IdentityMat( Length( M ) ); for i in [ 1 .. snf.rank ] do null[i][i] := n / GcdInt( n, snf.normal[i][i] ); od; nullM := null * snf.rowtrans; Assert( 1, ForAll( nullM, function ( v ) return v * M mod n = 0 * M[1]; end ) ); return nullM; end ); ############################################################################# ## #F RelationLatticeOfUnits( F, elms ) ## ## The input is a list of elements which generate a subgroup ## of the unit group of F. The function computes the relation lattice of ## . ## InstallGlobalFunction( RelationLatticeOfUnits, function( F, elms ) local exps,rels,l,record,rank,expsTorsion,rels1,rels2; # do a simple check if ForAll( elms, x -> x = x^0 ) then return IdentityMat( Length(elms) ); fi; # now compute an additive description record := ExponentsOfUnitsWithRank( F, elms ); exps := record.exps; rank := record.rank; # the first entry in the vectors of exps corresponds to # the torsion unit l := Length( exps[1] ); expsTorsion :=exps{[1..Length(exps)]}{[1]}; exps := exps{[1..Length(exps)]}{[2..l]}; # solve the first system mod rank rels1 := NullspaceModRank( expsTorsion, rank ); #if there are fundamental units solve second if l > 1 then rels2 := NullspaceIntMat( exps ); # get rels as the intersection of rels1 and rels2 rels := LatticeIntersection( rels1, rels2 ); else rels := rels1; fi; # format results if Length(rels) = 0 then return rels; fi; return NormalFormIntMat( rels, 2 ).normal; end ); ############################################################################# ## #F ExponentsOfFractionalIdealDescription( F, elms ) ## BindGlobal( "ExponentsOfFractionalIdealDescription", function( F, elms ) local base, flat, coef, exps, gens; # catch a trivial case if IsPrimeField(F) then return List( elms, x -> Norm( F, x ) ); fi; # determine exponents base := EquationOrderBasis( F ); coef := List( elms, x -> Coefficients( base, x ) ); exps := AL_FUNCS.ExponentsOfFractionalIdealDescription( F, coef ); # return exponents return exps; end ); ############################################################################# ## #F RelationLatticeModUnits( F, elms ) ## ## The function determines the relation lattice for modulo the unit ## group of F, i.e. relations rels between elms such that elms^rels is in ## the unitgroup of F. ## InstallGlobalFunction( RelationLatticeModUnits, function( F, elms ) local exps,rels,l; # do a simple check if ForAll( elms, x -> x = x^0 ) then return IdentityMat( Length(elms) ); fi; # now compute an additive description exps := ExponentsOfFractionalIdealDescription( F, elms ); # catch trivial case if Length(exps) = 0 then return IdentityMat(Length(elms)); fi; # compute the relations rels := NullspaceIntMat( exps ); # format results if Length(rels) = 0 then return rels; fi; return NormalFormIntMat( rels, 2 ).normal; end ); ############################################################################# ## #F RelationLatticeTF( F, elms ) ## ## The input is a list of elements which generate a free abelian subgroup ## of F. The function determines the relation lattice for , i.e. ## relations rels between such that elms^rels=1 ## InstallGlobalFunction( RelationLatticeTF, function( F, elms ) local rul,units,i,rl; # calculate the relation unit lattice rul := RelationLatticeModUnits( F, elms ); # calculate corresponding units units := []; for i in [1..Length(rul)] do Add( units, MappedVector( rul[i], elms )); od; # calculate the relations between the units rl := RelationLatticeOfTFUnits( F, units ); # catch trivial case if Length( rl ) = 0 then return []; fi; return NormalFormIntMat( rl * rul, 2).normal; end ); ############################################################################# ## #F RelationLatticePol( F, elms ) #M RelationLattice( F, elms ) ## ## The input is a list of elements in F. ## The function determines the relation lattice for , i.e. ## relations rels between such that elms^rels=1 ## BindGlobal( "RelationLatticePol", function( F, elms ) local rul,units,i,rl,F2,x; if DegreeOverPrimeField(F)=1 then x:=Indeterminate(Rationals); F2:=FieldByPolynomial(x^2-2); else F2:=F; fi; # calculate the relation unit lattice rul := RelationLatticeModUnits( F2, elms ); # calculate corresponding units units := []; for i in [1..Length(rul)] do Add( units, MappedVector( rul[i], elms )); od; # calculate the relations between the units rl := RelationLatticeOfUnits( F2, units ); # catch trivial case if Length( rl ) = 0 then return []; fi; return NormalFormIntMat( rl * rul, 2).normal; end ); BindGlobal( "RelationLatticeMat", function( F, elms ) local rul,units,i,rl,F2,x,c,elms2; if DegreeOverPrimeField(F)=1 then x:=Indeterminate(Rationals); c:=x^2-2; F2:=FieldByPolynomial(c); elms2 := List( elms, x-> x[1][1]*One(F2) ); return RelationLatticePol( F2, elms2 ); else F2:=F; fi; # calculate the relation unit lattice rul := RelationLatticeModUnits( F2, elms ); # calculate corresponding units units := []; for i in [1..Length(rul)] do Add( units, MappedVector( rul[i], elms )); od; # calculate the relations between the units rl := RelationLatticeOfUnits( F2, units ); # catch trivial case if Length( rl ) = 0 then return []; fi; return NormalFormIntMat( rl * rul, 2).normal; end ); InstallMethod( RelationLattice, "for fields by polynomial", true, [IsNumberField and IsAlgebraicExtension, IsCollection], 0, function( F, elms ) return RelationLatticePol( F, elms ); end); InstallMethod( RelationLattice, "for matrix fields", true, [IsNumberFieldByMatrices, IsCollection], 0, function( F, elms ) return RelationLatticeMat( F, elms ); end); alnuth-4.0.0/gap/setup.gd0000644000000000000000000000264615201210200012146 0ustar00############################################################################# ## #W setup.gd Alnuth - ALgebraic NUmber THeory Andreas Distler ## ############################################################################# ## #F SetPariStackSize(size) ## DeclareGlobalFunction("SetPariStackSize"); ############################################################################# ## #F SetAlnuthExternalExecutable(path) ## DeclareGlobalFunction("SetAlnuthExternalExecutable"); ############################################################################# ## #F AL_SuitablePariExecutable(path) ## DeclareGlobalFunction("AL_SuitablePariExecutable"); ############################################################################# ## #F PariVersion(path) ## DeclareGlobalFunction("PariVersion"); ############################################################################# ## #F SetAlnuthExternalExecutablePermanently(path) ## ## Deprecated wrapper around setting the persistent executable preference. ## DeclareGlobalFunction("SetAlnuthExternalExecutablePermanently"); ############################################################################# ## #F RestoreAlnuthExternalExecutablePermanently() ## ## Deprecated wrapper around restoring the default executable preference. ## DeclareGlobalFunction("RestoreAlnuthExternalExecutablePermanently"); ############################################################################# ## #E alnuth-4.0.0/gap/setup.gi0000644000000000000000000001021115201210200012136 0ustar00############################################################################# ## #W setup.gi Alnuth - ALgebraic NUmber THeory Andreas Distler ## BindGlobal("AL_InfoDeprecatedWrapper", function(name, replacement) Info(InfoObsolete, 1, "The function `", name, "` is deprecated. Use ", replacement, " instead."); end); ############################################################################# ## #F SetPariStackSize( size ) ## InstallGlobalFunction( SetPariStackSize, function( size ) if not IsPosInt( size ) then ErrorNoReturn(", the amount of memory in MB, must be a positive integer"); fi; SetUserPreference("alnuth", "PariStackSize", size); end ); ############################################################################# ## #F SetAlnuthExternalExecutable( path ) ## InstallGlobalFunction( SetAlnuthExternalExecutable, function( path ) AL_InfoDeprecatedWrapper( "SetAlnuthExternalExecutable", "SetUserPreference(\"alnuth\", \"PariGpPath\", path)" ); if not AL_SuitablePariExecutable(path) then ErrorNoReturn( " must name a usable PARI/GP executable in version 2.5 or higher" ); fi; SetUserPreference("alnuth", "PariGpPath", path); return UserPreference("alnuth", "PariGpPath"); end ); ############################################################################# ## #F AL_SuitablePariExecutable( path ) ## InstallGlobalFunction( AL_SuitablePariExecutable, function( path ) local resolved, version; if path = "" then return true; fi; resolved := AL_ResolvePariGpPath(path); if resolved = fail then Info(InfoWarning, 1, path, " is not an executable file."); return false; fi; version := ""; Process(DirectoryCurrent(), resolved, InputTextNone(), OutputTextString(version, false), [ "--version-short" ]); version := Chomp(version); if version = "" then Info(InfoWarning, 1, path, " does not seem to be an executable for PARI/GP."); return false; fi; if not CompareVersionNumbers(version, "2.5") then Info( InfoWarning, 1, path, " seems to be an executable for PARI/GP Version ", version, ", but Alnuth needs PARI/GP Version 2.5 or higher." ); return false; fi; return true; end ); ############################################################################# ## #F PariVersion( ) ## InstallGlobalFunction( PariVersion, function( ) local exe, str; exe := AL_CurrentPariGpPath(); if exe <> fail then # use the command line option to obtain version number of PARI/GP str := ""; Process( DirectoryCurrent( ), exe, InputTextNone( ), OutputTextString( str, false ), [ "--version-short" ] ); Print( str ); fi; end ); ############################################################################# ## #F SetAlnuthExternalExecutablePermanently( path ) ## InstallGlobalFunction( SetAlnuthExternalExecutablePermanently, function( path ) AL_InfoDeprecatedWrapper( "SetAlnuthExternalExecutablePermanently", "SetUserPreference(\"alnuth\", \"PariGpPath\", path); WriteGapIniFile()" ); if not AL_SuitablePariExecutable(path) then ErrorNoReturn( " must name a usable PARI/GP executable in version 2.5 or higher" ); fi; SetUserPreference("alnuth", "PariGpPath", path); WriteGapIniFile(); return UserPreference("alnuth", "PariGpPath"); end ); ############################################################################# ## #F RestoreAlnuthExternalExecutablePermanently( ) ## InstallGlobalFunction( RestoreAlnuthExternalExecutablePermanently, function( ) AL_InfoDeprecatedWrapper( "RestoreAlnuthExternalExecutablePermanently", "SetUserPreference(\"alnuth\", \"PariGpPath\", AL_DefaultPariGpPath()); WriteGapIniFile()" ); SetUserPreference("alnuth", "PariGpPath", AL_DefaultPariGpPath()); WriteGapIniFile(); return UserPreference("alnuth", "PariGpPath"); end ); ############################################################################# ## #E alnuth-4.0.0/gap/unithom.gi0000644000000000000000000000131515201210200012466 0ustar00############################################################################# ## #W unithom.gi Alnuth - ALgebraic NUmber THeory Bettina Eick #W Andreas Distler ## ############################################################################# ## #F Images under unit group homs ## InstallMethod( ImagesRepresentative, "for unit groups", FamSourceEqFamElm, [IsGroupGeneralMappingByImages and IsUnitGroupIsomorphism, IsMultiplicativeElementWithInverse], 0, function( nat, h ) local F, H, e; F := FieldOfUnitGroup( Source( nat ) ); H := Range( nat ); e := ExponentsOfUnits( F, [h] )[1]; return MappedVector( e, Pcp(H) ); end); alnuth-4.0.0/gap/userpref.gi0000644000000000000000000001265115201210200012643 0ustar00############################################################################# ## #W userpref.gi Alnuth - ALgebraic NUmber THeory ## BindGlobal("AL_DefaultPariGpPath", function() local f; f := Filename(DirectoriesSystemPrograms(), "gp"); if f = fail then return ""; fi; return f; end); BindGlobal("AL_ResolvePariGpPath", function(path) local resolved; if not IsString(path) or path = "" then return fail; fi; resolved := Filename(DirectoriesSystemPrograms(), path); if resolved <> fail and IsExecutableFile(resolved) then return resolved; fi; if IsExecutableFile(path) then return path; fi; return fail; end); BindGlobal("AL_SuitablePariGpPath", function(path) local resolved, str, version; if path = "" then return true; fi; resolved := AL_ResolvePariGpPath(path); if resolved = fail then return false; fi; str := ""; Process(DirectoryCurrent(), resolved, InputTextNone(), OutputTextString(str, false), [ "--version-short" ]); if str = "" then return false; fi; version := Chomp(str); return CompareVersionNumbers(version, "2.5"); end); BindGlobal("AL_WarnAboutObsoleteGlobals", function() local obsolete, names, name; names := [ "AL_EXECUTABLE", "AL_PATH", "AL_OPTIONS", "AL_STACKSIZE", "PRIM_TEST" ]; obsolete := []; for name in names do if IsBoundGlobal(name) then Add(obsolete, name); fi; od; if Length(obsolete) > 0 then Info(InfoWarning, 1, "Alnuth no longer supports the global variables ", JoinStringsWithSeparator(obsolete, ", "), ". Use SetUserPreference(\"alnuth\", ...) instead; see the manual ", "for the new preference names."); fi; return obsolete; end); BindGlobal("AL_CurrentPariGpPath", function() local path; path := AL_ResolvePariGpPath(UserPreference("alnuth", "PariGpPath")); if path <> fail then return path; fi; if IsBoundGlobal("AL_EXECUTABLE") and AL_SuitablePariGpPath(VALUE_GLOBAL("AL_EXECUTABLE")) then return AL_ResolvePariGpPath(VALUE_GLOBAL("AL_EXECUTABLE")); fi; return fail; end); BindGlobal("AL_CurrentPariGpOptions", function() local options; options := UserPreference("alnuth", "PariGpOptions"); if IsList(options) and ForAll(options, IsString) then return options; fi; if IsBoundGlobal("AL_OPTIONS") and IsList(VALUE_GLOBAL("AL_OPTIONS")) and ForAll(VALUE_GLOBAL("AL_OPTIONS"), IsString) then return VALUE_GLOBAL("AL_OPTIONS"); fi; return [ "-f", "-q" ]; end); BindGlobal("AL_LegacyStackSize", function() local val, digits; if not IsBoundGlobal("AL_STACKSIZE") then return fail; fi; val := VALUE_GLOBAL("AL_STACKSIZE"); if IsPosInt(val) then return val; fi; if IsString(val) and Length(val) >= 4 and val[1] = '-' and val[2] = 's' and val[Length(val)] = 'M' then digits := val{[3 .. Length(val) - 1]}; if ForAll(digits, ch -> ch in "0123456789") then return Int(digits); fi; fi; return fail; end); BindGlobal("AL_CurrentPariStackSize", function() local size; size := UserPreference("alnuth", "PariStackSize"); if IsPosInt(size) then return size; fi; size := AL_LegacyStackSize(); if size <> fail then return size; fi; return 128; end); BindGlobal("AL_CurrentPrimitiveElementTrials", function() local trials; trials := UserPreference("alnuth", "PrimitiveElementTrials"); if IsPosInt(trials) then return trials; fi; if IsBoundGlobal("PRIM_TEST") and IsPosInt(VALUE_GLOBAL("PRIM_TEST")) then return VALUE_GLOBAL("PRIM_TEST"); fi; return 20; end); DeclareUserPreference(rec( package := "alnuth", name := "PariGpPath", description := [ """is the command or path for the PARI/GP executable used by Alnuth. The default searches for a program named gp in DirectoriesSystemPrograms(). Set this preference if PARI/GP is installed elsewhere or should be invoked with a different executable name. An empty string disables the PARI/GP backend until a usable executable is configured. """], default := AL_DefaultPariGpPath, check := x -> IsString(x) and AL_SuitablePariGpPath(x) )); DeclareUserPreference(rec( package := "alnuth", name := "PariGpOptions", description := [ """are the command line options passed to PARI/GP for every Alnuth call. The default disables user startup files and the interactive prompt. In normal use there is no need to change this preference. """], default := ["-f", "-q"], check := x -> IsList(x) and ForAll(x, IsString) )); DeclareUserPreference(rec( package := "alnuth", name := "PariStackSize", description := [ """is the stack size in megabytes requested from PARI/GP. The value is translated into a -s... command line argument when Alnuth starts PARI/GP. It must be a positive integer. """], default := 128, check := IsPosInt )); DeclareUserPreference(rec( package := "alnuth", name := "PrimitiveElementTrials", description := [ """is the number of random trials used when Alnuth searches for a primitive element with a small defining polynomial. Larger values may produce nicer defining polynomials at the cost of more work. """], default := 20, check := IsPosInt )); alnuth-4.0.0/gp/0000755000000000000000000000000015201210200010321 5ustar00alnuth-4.0.0/gp/decompra.gp0000644000000000000000000000077215201210200012451 0ustar00bnf = bnfinit(f,1); n = poldegree(f); un = lift(concat([bnf.tu[2]],bnf.fu)); r = #un; p2v(n,b) = vector(n,j,polcoeff(b,j-1)); \\print units print("[[ "); for(i=1,#un, print(p2v(n,un[i]),",")); print("],\n"); \\print exponents print("[ "); { for(i=1,#elms-1, c = bnfisunit(bnf, Polrev(elms[i])); if (#c==0, error("element must be a unit")); c = vector(r,j,if(j==1,lift(c[r]),c[j-1])); print(c,","); ); } print("],\n"); \\ get the rank of the group rank=bnf.tu[1]; print(rank,"];"); alnuth-4.0.0/gp/fracidea.gp0000644000000000000000000000117615201210200012414 0ustar00\\ calculates the exponentsvectors of the fractional ideals \\ generated by elms corresponding to the prime ideals \\ arrissing in the factorization nf = nfinit(f); \\ factorize all ideals generated by the elements of a and collect all \\ primideals which occur primId=eval(Set(concat(vector(#elms-1,i,idealfactor(nf,Polrev(elms[i]))[,1])))); \\ write for every element in a the exponentfactor corresponding to the \\ ideals in primId in a row \\ so we will have a matrix mat = matrix(#elms-1,#primId,i,j,idealval(nf,Polrev(elms[i]),primId[j])); print("[ "); { for(i=1,matsize(mat)[1], print(mat[i,],","); ); print("];"); } alnuth-4.0.0/gp/maxord.gp0000644000000000000000000000025615201210200012146 0ustar00\\ compute basis of maximal order b = nfbasis( f ); p2v(n,b)=vector(n,j,polcoeff(b,j-1)); \\ print result print("[ "); for(i=1,#b, print(p2v(#b,b[i]),",")); print("];"); alnuth-4.0.0/gp/norm.gp0000644000000000000000000000051015201210200011620 0ustar00bnf = bnfinit(f,1); n = poldegree(f); un = lift(concat([bnf.tu[2]],bnf.fu)); nr = bnfisintnorm( bnf, nrm); p2v(n,b) = vector(n,j,polcoeff(b,j-1)); \\ print units print("[[ "); for(i=1,#un, print(p2v(n,un[i]),",")); print("],\n"); \\ print norm elements print(" [ "); for (i=1, #nr, print(p2v(n,nr[i]),",\n")); print("]];"); alnuth-4.0.0/gp/polyfactors.gp0000644000000000000000000000070315201210200013216 0ustar00\\ compute factors of poly defined by coeffs over Q_f f = subst(f,variable(f),'varA); pol = Pol(vector(#coeffs-1,i,Polrev(coeffs[i],'varA))); n = poldegree(f); gettime(); fac = lift(nffactor(f, pol )); zeit = gettime(); p2v(n,b)=vector(n,j,polcoeff(b,j-1)); f2v(n,v)=vector(#v,i,p2v(n,v[i])); \\ print result print("[ "); { for(i=1,#fac[,1], for(j=1,fac[i,2], print(f2v(n, Vec(fac[i,1]) ),","); ) ); } print1(zeit); print("];"); alnuth-4.0.0/gp/units.gp0000644000000000000000000000031115201210200012006 0ustar00bnf = bnfinit(f,1); n = poldegree(f); un = lift(concat([bnf.tu[2]],bnf.fu)); p2v(n,b) = vector(n,j,polcoeff(b,j-1)); \\ print units print("[ "); for(i=1,#un, print(p2v(n,un[i]),",")); print("];\n"); alnuth-4.0.0/init.g0000644000000000000000000000071315201210200011027 0ustar00############################################################################# ## #W init.g Alnuth - ALgebraic NUmber THeory Bettina Eick #W Andreas Distler ## ############################################################################# ## #R read .gd files ## ReadPackage("alnuth", "gap/setup.gd"); ReadPackage("alnuth", "gap/factors.gd"); ReadPackage("alnuth", "gap/field.gd"); alnuth-4.0.0/makedoc.g0000644000000000000000000000104715201210200011470 0ustar00## this creates the documentation, needs: GAPDoc and AutoDoc packages, pdflatex ## ## Call this with GAP from within the package directory. ## if fail = LoadPackage("AutoDoc", ">= 2022.07.10") then Error("AutoDoc 2022.07.10 or newer is required"); fi; AutoDoc(rec( autodoc := true, extract_examples := true, scaffold := rec( TitlePage := false, includes := [ "intro.xml", "fields.xml", "example.xml", "install.xml", ], bib := "manual.bib", ), )); alnuth-4.0.0/read.g0000644000000000000000000000200015201210200010766 0ustar00############################################################################# ## #W read.g Alnuth - ALgebraic NUmber THeory Bettina Eick ## Andreas Distler ## ############################################################################# ## #R read files ## ReadPackage("alnuth", "gap/userpref.gi"); AL_WarnAboutObsoleteGlobals(); ReadPackage("alnuth", "gap/setup.gi"); ReadPackage("alnuth", "gap/pari.gi"); ReadPackage("alnuth", "gap/factors.gi"); ReadPackage("alnuth", "gap/matfield.gi"); ReadPackage("alnuth", "gap/polfield.gi"); ReadPackage("alnuth", "gap/field.gi"); ReadPackage("alnuth", "gap/unithom.gi"); ReadPackage("alnuth", "gap/matunits.gi"); ReadPackage("alnuth", "gap/rels.gi"); ReadPackage("alnuth", "gap/present.gi"); ReadPackage("alnuth", "gap/isom.gi"); ReadPackage("alnuth", "gap/rationals.gi"); ReadPackage("alnuth", "exam/unimod.gi"); ReadPackage("alnuth", "exam/rationals.gi"); ReadPackage("alnuth", "exam/fields.gi"); alnuth-4.0.0/tst/0000755000000000000000000000000015201210200010525 5ustar00alnuth-4.0.0/tst/ALNUTH.tst0000644000000000000000000000661715201210200012266 0ustar00gap> START_TEST("Installation test of Alnuth package"); gap> mats := ExamUnimod( 1 );; gap> F := FieldByMatrices( mats ); gap> DegreeOverPrimeField( F ); 4 gap> IsFinite( F ); false gap> EquationOrderBasis( F ); Basis( , [ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 11, 11, 22, -22 ], [ 33, -33, -66, 132 ], [ 22, -22, -22, 55 ], [ 20, -24, -38, 80 ] ], [ [ 528, -198, -132, 660 ], [ 462, -264, -660, 1848 ], [ 132, 132, 330, -198 ], [ 192, -72, -180, 702 ] ], [ [ 9570, -594, 2508, 7788 ], [ 18810, -16038, -28116, 66528 ], [ 9108, -5412, -5544, 16830 ], [ 9816, -8400, -13740, 32532 ] ] ] ) # testing maxord.gp gap> IsIntegerOfNumberField( F, mats[1] ); true gap> MaximalOrderBasis( F );; # testing units.gp gap> UnitGroup( F ); gap> IsCyclotomicField( F ); false # testing fracidea.gp and decompra.gp gap> IsomorphismPcpGroup( F, mats{[2..5]} ); [ [ [ 57641556673, -51250063536, -73214376480, 161071628256 ], [ 21964312944, 28355806081, 43928625888, 0 ], [ -14642875296, 43928625888, 64962994321, -80535814128 ], [ 0, 29285750592, 43928625888, -37537132751 ] ], [ [ 13, 0, -21, 0 ], [ -42, 97, 105, -231 ], [ -21, 0, 34, 0 ], [ -21, 21, 42, -50 ] ], [ [ 6113341760402965, -3032143586011050, -4159967272068153, 14002438585824810 ], [ 10588511869480164, -5251666322974043, -7205040811308855, 24252552936374367 ], [ 3778184141734557, -1873864241780610, -2570850209123252, 8653736311352880 ], [ 5051133104082267, -2505235179386847, -3437063756709534, 11569395035183716 ] ] ] -> [ g1, g2, g3 ] gap> RelationLatticeOfUnits( F, mats ); [ [ 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 2, -2 ], [ 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2 ], [ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0 ], [ 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 1, -4 ], [ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2 ], [ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, -3 ] ] # testing polyfactors.gp gap> pol := UnivariatePolynomial( Rationals, [0,0,8,0,8,2,0,2] ); 2*x^7+2*x^5+8*x^4+8*x^2 gap> f := UnivariatePolynomial( Rationals, [-4,0,0,1] ); x^3-4 gap> L := FieldByPolynomial( f ); gap> FactorsPolynomialAlgExt( L, pol ); [ !2*x_1, x_1, x_1+a, x_1^2+!1, x_1^2+(-a)*x_1+a^2 ] gap> pol := UnivariatePolynomial( Rationals, [ 1, 3, 2, -1, 2, 3, 1 ] ); x^6+3*x^5+2*x^4-x^3+2*x^2+3*x+1 gap> f := UnivariatePolynomial( Rationals,[ 11/64, 59/16, -7/4, 1 ] ); x^3-7/4*x^2+59/16*x+11/64 gap> L := FieldByPolynomial( f ); gap> FactorsPolynomialAlgExt( L, pol ); [ x_1^2+x_1+(-a+1/4), x_1^2+(-a^2+3/2*a-21/16)*x_1+!1, x_1^2+(a^2-3/2*a+53/16)*x_1+(a^2-3/2*a+53/16) ] # testing norm.gp and fracidea.gp gap> pol := UnivariatePolynomial( Rationals, [ 1, 0, -1, 1 ] ); x^3-x^2+1 gap> L := FieldByPolynomial( pol ); gap> cosets := NormCosetsOfNumberField( L, 5 );; gap> [Norm(cosets[1]), Length(cosets)]; [ 5, 1 ] gap> ExponentsOfFractionalIdealDescription( L, cosets ); [ [ 1 ] ] gap> STOP_TEST( "ALNUTH.tst", 100000); alnuth-4.0.0/tst/alnuth01.tst0000644000000000000000000000157615201210200012726 0ustar00# Alnuth, chapter 2 # # DO NOT EDIT THIS FILE - EDIT EXAMPLES IN THE SOURCE INSTEAD! # # This file has been generated by AutoDoc. It contains examples extracted from # the package documentation. Each example is preceded by a comment which gives # the name of a GAPDoc XML file and a line range from which the example were # taken. Note that the XML file in turn may have been generated by AutoDoc # from some other input. # gap> START_TEST("alnuth01.tst"); # doc/fields.xml:251-262 gap> x := Indeterminate( Rationals, "x" );; gap> pol := 2*x^7+2*x^5+8*x^4+8*x^2; 2*x^7+2*x^5+8*x^4+8*x^2 gap> L := FieldByPolynomial( x^3-4 ); gap> y := Indeterminate( L, "y" );; gap> FactorsPolynomialAlgExt( L, pol ); [ !2*y, y, y+a, y^2+!1, y^2+(-a)*y+a^2 ] gap> FactorsPolynomialAlnuth( last[5] ); [ y^2+(-a)*y+a^2 ] # gap> STOP_TEST("alnuth01.tst", 1); alnuth-4.0.0/tst/alnuth02.tst0000644000000000000000000000574715201210200012733 0ustar00# Alnuth, chapter 3 # # DO NOT EDIT THIS FILE - EDIT EXAMPLES IN THE SOURCE INSTEAD! # # This file has been generated by AutoDoc. It contains examples extracted from # the package documentation. Each example is preceded by a comment which gives # the name of a GAPDoc XML file and a line range from which the example were # taken. Note that the XML file in turn may have been generated by AutoDoc # from some other input. # gap> START_TEST("alnuth02.tst"); # doc/example.xml:13-70 gap> m1 := [ [ 1, 0, 0, -7 ], > [ 7, 1, 0, -7 ], > [ 0, 7, 1, -7 ], > [ 0, 0, 7, -6 ] ];; # gap> m2 := [ [ 0, 0, -13, 14 ], > [ -1, 0, -13, 1 ], > [ 13, -1, -13, 1 ], > [ 0, 13, -14, 1 ] ];; # gap> F := FieldByMatricesNC( [m1, m2] ); # gap> DegreeOverPrimeField(F); 4 gap> PrimitiveElement(F); [ [ 0, -1, 1, 0 ], [ 0, -1, 0, 1 ], [ 0, -1, 0, 0 ], [ 1, -1, 0, 0 ] ] # gap> Basis(F); Basis( , [ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 0, 1, 0, 0 ], [ -1, 1, 1, 0 ], [ -1, 0, 1, 1 ], [ -1, 0, 0, 1 ] ], [ [ 0, 0, 1, 0 ], [ -1, 0, 1, 1 ], [ -1, -1, 1, 1 ], [ 0, -1, 0, 1 ] ], [ [ 0, 0, 0, 1 ], [ -1, 0, 0, 1 ], [ 0, -1, 0, 1 ], [ 0, 0, -1, 1 ] ] ] ) # gap> MaximalOrderBasis(F); Basis( , [ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 0, -1, 1, 0 ], [ 0, -1, 0, 1 ], [ 0, -1, 0, 0 ], [ 1, -1, 0, 0 ] ], [ [ 0, 0, 0, -1 ], [ 1, 0, 0, -1 ], [ 0, 1, 0, -1 ], [ 0, 0, 1, -1 ] ], [ [ -1, 1, 0, 0 ], [ -1, 0, 1, 0 ], [ -1, 0, 0, 1 ], [ -1, 0, 0, 0 ] ] ] ) # gap> U := UnitGroup(F); # gap> u := GeneratorsOfGroup( U );; # gap> nat := IsomorphismPcpGroup(U);; gap> H := Image(nat); Pcp-group with orders [ 10, 0 ] gap> ImageElm( nat, u[1] ); g1 gap> ImageElm( nat, u[2] ); g2 gap> ImageElm( nat, u[1]*u[2] ); g1*g2 gap> u[1] = PreImagesRepresentative(nat, GeneratorsOfGroup(H)[1] ); true # doc/example.xml:78-114 gap> g := UnivariatePolynomial( Rationals, [ 16, 64, -28, -4, 1 ] ); x^4-4*x^3-28*x^2+64*x+16 # gap> F := FieldByPolynomialNC(g); gap> PrimitiveElement(F); a gap> MaximalOrderBasis(F); Basis( , [ !1, 1/2*a, 1/4*a^2, 1/56*a^3+1/14*a^2+1/14*a-2/7 ] ) # gap> U := UnitGroup(F); # gap> natU := IsomorphismPcpGroup(U);; gap> elms := List( [1..10], x-> Random(F) );; # gap> PcpPresentationOfMultiplicativeSubgroup( F, elms ); Pcp-group with orders [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] # gap> isom := IsomorphismPcpGroup( F, elms );; gap> y := RandomGroupElement( elms );; gap> z := ImageElm( isom, y );; gap> y = PreImagesRepresentative( isom, z ); true # gap> FactorsPolynomialAlgExt( F, g ); [ x_1+(-a), x_1+(a-2), x_1+(-1/7*a^3+3/7*a^2+31/7*a-40/7), x_1+(1/7*a^3-3/7*a^2-31/7*a+26/7) ] # gap> STOP_TEST("alnuth02.tst", 1); alnuth-4.0.0/tst/examples.tst0000644000000000000000000002450115201210200013101 0ustar00gap> START_TEST("Testing examples from Alnuth"); # example 1 gap> F := ExampleMatField(1); gap> DegreeOverPrimeField(F); 4 gap> DefiningPolynomial(F); x^4+4*x^3-394*x^2-796*x+2101 gap> basis := Basis(F, > [ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], > [ [ -1, 1, 1, 0 ], [ 5, -5, -5, 11 ], [ 3, -4, -7, 11 ], [ 3, -3, -4, 7 ] ], > [ [ 9, -10, -13, 22 ], [ -12, 17, 21, -33 ], [ -11, 18, 28, -44 ], > [ -9, 13, 18, -28 ] ], > [ [ -3, 5, 7, -11 ], [ 7, -9, -13, 22 ], [ 6, -9, -13, 22 ], > [ 5, -7, -10, 17 ] ] ]);; gap> ForAll(BasisVectors(basis), mat-> IsIntegerOfNumberField(F, mat)); true gap> ForAll(BasisVectors(MaximalOrderBasis(F)), > mat-> ForAll( Coefficients( basis, mat ), IsInt)); true gap> ug := UnitGroup(F); gap> RelationLatticeOfUnits(F,GeneratorsOfGroup(ug)); [ [ 2, 0, 0, 0 ] ] # example 2 gap> F := ExampleMatField(2); gap> DegreeOverPrimeField(F); 4 gap> DefiningPolynomial(F); x^4-18*x^3-286*x^2+2178*x+14641 gap> basis := Basis(F, > [ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], > [ [ -1, 1, 1, 0 ], [ -4, -9, -5, 11 ], [ -6, -8, -7, 11 ], > [ -5, -8, -5, 11 ] ], > [ [ -9, -18, -13, 22 ], [ 15, 29, 21, -33 ], [ 25, 34, 28, -44 ], > [ 12, 19, 15, -22 ] ], > [ [ 6, 9, 7, -11 ], [ -11, -17, -13, 22 ], [ -12, -17, -13, 22 ], > [ -8, -12, -9, 16 ] ] ] );; gap> ForAll(BasisVectors(basis), mat-> IsIntegerOfNumberField(F, mat)); true gap> ForAll(BasisVectors(MaximalOrderBasis(F)), > mat-> ForAll( Coefficients( basis, mat ), IsInt)); true gap> ug := UnitGroup(F); gap> Size(ug); infinity # example 3 gap> F := ExampleMatField(3); gap> DegreeOverPrimeField(F); 4 gap> DefiningPolynomial(F); x^4+4*x^3-90*x^2-188*x+1669 gap> basis := Basis( F, > [ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], > [ [ -4, 6, 7, 0 ], [ -31, -60, -35, 77 ], [ -35, -48, -41, 66 ], > [ -34, -53, -33, 75 ] ], > [ [ -11, -15, -10, 22 ], [ 3, 3, 6, 0 ], [ 7, 10, 8, -11 ], > [ -3, -5, 0, 12 ] ], > [ [ 4, 10, 8, -11 ], [ -15, -27, -18, 33 ], [ -18, -25, -21, 33 ], > [ -13, -20, -14, 26 ] ] ] );; gap> ForAll(BasisVectors(basis), mat-> IsIntegerOfNumberField(F, mat)); true gap> ForAll(BasisVectors(MaximalOrderBasis(F)), > mat-> ForAll( Coefficients( basis, mat ), IsInt)); true gap> ug := UnitGroup(F); gap> Size(ug); infinity # example 4 gap> F := ExampleMatField(4); gap> DegreeOverPrimeField(F); 4 gap> DefiningPolynomial(F); x^4-8*x^3-550*x^2-1936*x-1331 gap> basis := Basis( F, > [ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], > [ [ 0, -1, 1, 0 ], [ -1, -4, -5, 11 ], [ -1, 4, 4, -11 ], [ -1, 1, 0, -2 ] ] > , > [ [ 0, 8, 9, -22 ], [ -2, 8, -1, -11 ], [ 3, -10, -5, 22 ], > [ 1, -5, -6, 15 ] ], > [ [ 1, -5, -3, 11 ], [ 1, -9, -7, 22 ], [ -2, 9, 7, -22 ], [ -1, 3, 2, -7 ] > ] ] );; gap> ForAll(BasisVectors(basis), mat-> IsIntegerOfNumberField(F, mat)); true gap> ForAll(BasisVectors(MaximalOrderBasis(F)), > mat-> ForAll( Coefficients( basis, mat ), IsInt)); true gap> ug := UnitGroup(F); gap> Size(ug); infinity # example 5 gap> F := ExampleMatField(5); gap> DegreeOverPrimeField(F); 4 gap> DefiningPolynomial(F); x^4+4*x^3-40*x^2-88*x+244 gap> basis := Basis( F, > [ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], > [ [ 148, 95, 101, -165 ], [ -257, -165, -175, 286 ], [ 96, 65, 67, -110 ], > [ -64, -40, -43, 70 ] ], > [ [ 3549, 2310, 2437, -3982 ], [ -6147, -4001, -4221, 6897 ], > [ 2195, 1430, 1508, -2464 ], [ -1560, -1015, -1071, 1750 ] ], > [ [ 2999, 1953, 2060, -3366 ], [ -5195, -3383, -3568, 5830 ], > [ 1853, 1206, 1272, -2079 ], [ -1319, -859, -906, 1480 ] ] ] );; gap> ForAll(BasisVectors(basis), mat-> IsIntegerOfNumberField(F, mat)); true gap> ForAll(BasisVectors(MaximalOrderBasis(F)), > mat-> ForAll( Coefficients( basis, mat ), IsInt)); true gap> ug := UnitGroup(F); gap> Size(ug); infinity # example 6 gap> F := ExampleMatField(6); gap> DegreeOverPrimeField(F); 4 gap> DefiningPolynomial(F); x^4-34*x^2+49 gap> basis := Basis( F, > [ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], > [ [ -6, -2, -3, 0 ], [ 15, 18, 15, -33 ], [ 11, 16, 7, -22 ], > [ 14, 21, 13, -37 ] ], > [ [ -2, -11, -4, 22 ], [ -27, -36, -24, 66 ], [ -15, -22, -9, 33 ], > [ -31, -47, -26, 83 ] ], > [ [ 29, 55, 26, -99 ], [ 82, 118, 68, -209 ], [ 37, 55, 25, -88 ], > [ 102, 155, 82, -270 ] ] ] );; gap> ForAll(BasisVectors(basis), mat-> IsIntegerOfNumberField(F, mat)); true gap> ForAll(BasisVectors(MaximalOrderBasis(F)), > mat-> ForAll( Coefficients( basis, mat ), IsInt)); true gap> ug := UnitGroup(F); gap> Size(ug); infinity # example 7 gap> F := ExampleMatField(7); gap> DegreeOverPrimeField(F); 4 gap> DefiningPolynomial(F); x^4-9118*x^3+39843*x^2-9118*x+1 gap> basis := Basis( F, > [ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], > [ [ -16591/25, -4204252/75, -1139984/125, 933941/25 ], > [ 127557/100, 2685042/25, 2183967/125, -3578691/50 ], > [ 160, 39760/3, 2155, -35325/4 ], > [ 47976/25, 4036224/25, 3282924/125, -2689776/25 ] ], > [ [ -172958/5, -227281196/75, -61682584/125, 50500116/25 ], > [ 1325391/20, 145139966/25, 118170042/125, -193493841/50 ], > [ 32695/4, 2148280/3, 116606, -1909325/4 ], > [ 498078/5, 218173052/25, 177632124/125, -145429076/25 ] ], > [ [ -1572318/25, -137719084/25, -22425524/25, 91800234/25 ], > [ 3012234/25, 263839042/25, 42962292/25, -703475043/100 ], > [ 29725/2, 1301735, 211968, -867705 ], > [ 4527973/25, 1189801372/75, 64580524/25, -264364324/25 ] ] ] );; gap> ForAll(BasisVectors(basis), mat-> IsIntegerOfNumberField(F, mat)); true gap> ForAll(BasisVectors(MaximalOrderBasis(F)), > mat-> ForAll( Coefficients( basis, mat ), IsInt)); true gap> ug := UnitGroup(F); gap> Size(ug); infinity # example 8 gap> F := ExampleMatField(8); gap> DegreeOverPrimeField(F); 4 gap> DefiningPolynomial(F); x^4-2693461698*x^3+915480803*x^2-183198*x+1 gap> basis := Basis( F, > [ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], > [ [ 350938712911/294, 1500326345/2, 22032258785/49, 12381906785/28 ], > [ -321440829383/245, -824530541, -48432850131/98, -6804693081/14 ], > [ -1895565335102/441, -8103883972/3, -238010709389/147, -33439903349/21 > ], [ 423681072037/105, 2535838285, 10639636697/7, 2989684151/2 ] ], > [ [ 4296545420323143572/147, 1800113708794452685/98, 539482234887109565/49, > 1061142474502207065/98 ], > [ -23612415601847672819/735, -989283921486886210/49, > -592963764782450491/49, -1166338752161250809/98 ], > [ -232074212936617048906/2205, -9723159685193494472/147, > -5827933970510514158/147, -1910556664266896923/49 ], > [ 3458087129618678238/35, 434648032727934931/7, 260522311384860087/7, > 256219085901701948/7 ] ], > [ [ -359040394192145849/98, -225639486739431445/98, > -270490678428289125/196, -266022798598028295/196 ], > [ 2959753770581068096/735, 124004119958269647/49, 74326437761644025/49, > 73098737073176767/49 ], > [ 3232209784961308454/245, 1218772319845216676/147, > 243505363658010411/49, 239483218757603493/49 ], > [ -433462064765870443/35, -54481979964874243/7, -65311563750512469/14, > -64232767911560593/14 ] ] ] );; gap> ForAll(BasisVectors(basis), mat-> IsIntegerOfNumberField(F, mat)); true gap> ForAll(BasisVectors(MaximalOrderBasis(F)), > mat-> ForAll( Coefficients( basis, mat ), IsInt)); true gap> ug := UnitGroup(F); gap> Size(ug); infinity # example 9 gap> F := ExampleMatField(9); gap> DegreeOverPrimeField(F); 4 gap> DefiningPolynomial(F); x^4-50691194176*x^3+13505436470112846*x^2-5255736770373376*x+1 gap> basis := Basis( F, > [ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], > [ > [ 851504247972/229, -1937936674346/229, -603837909603/229, > 1577056081004/229 ], > [ 589771389309/458, -671133447401/229, -209117560764/229, > 546155850465/229 ], > [ -314679082561/229, 2148551139301/687, 223154956025/229, > -582817030683/229 ], > [ 260652277735/458, -889841242682/687, -92421808448/229, > 241378880370/229 ] ], > [ > [ 479595170949875358661/229, -3274539508376250773645/687, > -340102839979551252617/229, 888252880017494540431/229 ], > [ 166089147656173639110/229, -378003163829060810519/229, > -235562800592787808287/458, 307611862433136447321/229 ], > [ -177237322072221156244/229, 1210126057694719416842/687, > 125687079941754647751/229, -328259272238145784528/229 ], > [ 73403753002708619935/229, -167059987151745488349/229, > -104107907570323308205/458, 135950274239559405325/229 ] ], > [ > [ -168577244280151879851/229, 1150997507992512356939/687, > 119545823245159623469/229, -312220038497649730859/229 ], > [ -116760353367458396067/458, 265735501720229196779/458, > 41400049650414034161/229, -216250551372975665307/458 ], > [ 62298749337598187302/229, -425358152864100770527/687, > -44178888490966125184/229, 115382820501957885815/229 ], > [ -51602698069330099209/458, 352328555032045040059/1374, > 18296915011903387616/229, -95572782957948415617/458 ] ] ] );; gap> ForAll(BasisVectors(basis), mat-> IsIntegerOfNumberField(F, mat)); true gap> ForAll(BasisVectors(MaximalOrderBasis(F)), > mat-> ForAll( Coefficients( basis, mat ), IsInt)); true gap> ug := UnitGroup(F); gap> Size(ug); infinity # no more examples gap> F := ExampleMatField(10); fail gap> STOP_TEST( "examples.tst", 10000000); alnuth-4.0.0/tst/polynome.tst0000644000000000000000000000366115201210200013131 0ustar00# this needs RadiRoot to be loaded gap> START_TEST("Factorisation of polynomials using Alnuth"); # some splitting fields gap> pol := UnivariatePolynomial( Rationals, [ -1, 0, 0, 0, 1, 0, 1 ] );; gap> SplittingField( pol ); gap> pol := UnivariatePolynomial( Rationals, [ 4, -1, 0, 6, -1, -2, 2, 1 ] );; gap> SplittingField( pol ); gap> pol := UnivariatePolynomial( Rationals, [ 4, 0, 4, 0, -3, 0, -1, 0, 1 ] );; gap> SplittingField( pol ); gap> pol := UnivariatePolynomial(Rationals, > [ -3, 1, -1, 2, 1, 3, -2, -1, 0, 1 ]);; gap> SplittingField( pol ); gap> pol := UnivariatePolynomial(Rationals, > [ 47, 0, 103, 0, 41, 0, 7, 0, -2, 0, 1 ]);; gap> SplittingField( pol ); # example from email to Bill Allombert (10/05/11) gap> pol := UnivariatePolynomial( Rationals, [ 1, 2, 2, 2, 2, 1, 1 ] ); x^6+x^5+2*x^4+2*x^3+2*x^2+2*x+1 gap> f := UnivariatePolynomial( Rationals, [ 1, 1, 1 ] ); x^2+x+1 gap> K := SplittingField( pol ); gap> FactorsPolynomialAlgExt( K, f ); [ x_1^2+x_1+!1 ] # example from Bill Allombert (10/05/11) gap> pol := UnivariatePolynomial( Rationals, [ 20736, 0, 4147200, 0, 60632064, > 0, 347286528, 0, 1078555392, 0, 2069549568, 0, 2613917952, 0, 2241209088, 0, > 1318874976, 0, 530669952, 0, 143684928, 0, 25510464, 0, 2872752, 0, 195936, 0, > 7536, 0, 144, 0, 1 ] );; gap> L := FieldByPolynomial( pol ); gap> facs := FactorsPolynomialAlgExt( L, pol );; gap> Collected(List(facs, Degree)); [ [ 1, 32 ] ] gap> Product(facs) = AlgExtEmbeddedPol(L, pol); true # gap> STOP_TEST( "polynome.tst", 10000000); alnuth-4.0.0/tst/testall.g0000644000000000000000000000044415201210200012347 0ustar00LoadPackage("alnuth"); LoadPackage("radiroot"); # ensure name of indeterminate matches the manual x := Indeterminate(Rationals, "x"); dirs := DirectoriesPackageLibrary( "alnuth", "tst" ); TestDirectory(dirs, rec(exitGAP := true, testOptions:=rec(compareFunction := "uptowhitespace"))); alnuth-4.0.0/tst/testinstall.g0000644000000000000000000000046415201210200013247 0ustar00LoadPackage("alnuth"); # ensure name of indeterminate matches the manual x := Indeterminate(Rationals, "x"); dirs := DirectoriesPackageLibrary( "alnuth", "tst" ); tests := [ "ALNUTH.tst", "userprefs.tst", ]; tests := List(tests, f -> Filename(dirs,f)); TestDirectory(tests, rec(exitGAP := false)); alnuth-4.0.0/tst/userprefs.tst0000644000000000000000000000472515201210200013307 0ustar00gap> START_TEST("userprefs.tst"); gap> pari_path:=UserPreference("alnuth", "PariGpPath");; gap> IsString(pari_path); true gap> UserPreference("alnuth", "PariGpOptions"); [ "-f", "-q" ] gap> UserPreference("alnuth", "PariStackSize"); 128 gap> UserPreference("alnuth", "PrimitiveElementTrials"); 20 gap> tmpdir := DirectoryTemporary();; gap> oldroot := GAPInfo.UserGapRoot;; gap> oldobsolete := InfoLevel(InfoObsolete);; gap> oldinfo := InfoLevel(InfoWarning);; gap> GAPInfo.UserGapRoot := tmpdir![1];; gap> SetInfoLevel(InfoObsolete, 1);; gap> SetInfoLevel(InfoWarning, 0);; gap> path := SetAlnuthExternalExecutable("gp");; #I The function `SetAlnuthExternalExecutable` is deprecated. Use SetUserPreference("alnuth", "PariGpPath", path) instead. gap> IsString(path); true gap> path := SetAlnuthExternalExecutablePermanently("gp");; #I The function `SetAlnuthExternalExecutablePermanently` is deprecated. Use SetUserPreference("alnuth", "PariGpPath", path); WriteGapIniFile() instead. gap> IsString(path); true gap> IsExistingFile(Filename(tmpdir, "gap.ini")); true gap> oldparipath := UserPreference("alnuth", "PariGpPath");; gap> oldpariopts := UserPreference("alnuth", "PariGpOptions");; gap> oldparistack := UserPreference("alnuth", "PariStackSize");; gap> oldprimtrials := UserPreference("alnuth", "PrimitiveElementTrials");; gap> SetUserPreference("alnuth", "PariGpPath", "");; gap> AL_EXECUTABLE := "gp";; gap> IsString(AL_CurrentPariGpPath()); true gap> SetUserPreference("alnuth", "PariGpOptions", 1);; gap> AL_OPTIONS := [ "-f" ];; gap> AL_CurrentPariGpOptions(); [ "-f" ] gap> SetUserPreference("alnuth", "PariStackSize", 0);; gap> AL_STACKSIZE := "-s256M";; gap> AL_CurrentPariStackSize(); 256 gap> SetUserPreference("alnuth", "PrimitiveElementTrials", 0);; gap> PRIM_TEST := 7;; gap> AL_CurrentPrimitiveElementTrials(); 7 gap> AL_OPTIONS := [ "-f" ];; gap> AL_STACKSIZE := "-s128M";; gap> AL_WarnAboutObsoleteGlobals(); [ "AL_EXECUTABLE", "AL_OPTIONS", "AL_STACKSIZE", "PRIM_TEST" ] gap> SetUserPreference("alnuth", "PariGpPath", oldparipath);; gap> SetUserPreference("alnuth", "PariGpOptions", oldpariopts);; gap> SetUserPreference("alnuth", "PariStackSize", oldparistack);; gap> SetUserPreference("alnuth", "PrimitiveElementTrials", oldprimtrials);; gap> SetInfoLevel(InfoObsolete, oldobsolete);; gap> SetInfoLevel(InfoWarning, oldinfo);; gap> GAPInfo.UserGapRoot := oldroot;; gap> Unbind(AL_EXECUTABLE); Unbind(AL_OPTIONS); Unbind(AL_STACKSIZE); Unbind(PRIM_TEST); gap> STOP_TEST("userprefs.tst", 1);