  
  
                                    [1X Alnuth [101X
  
  
        [1X ALgebraic NUmber THeory and an interface to PARI/GP and OSCAR [101X
  
  
                                     4.0.2
  
  
                                  21 June 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
  
  
  
            [33X[0;10Y[13XNote:[113X  PARI/GP  is  [13Xnot[113X  part of this package. It can be
            obtained  from  [7Xhttps://pari.math.u-bordeaux.fr/[107X. If you
            use GAP via OSCAR, then OSCAR will automatically be used
            instead of PARI/GP.[133X
  
  
  
  
  -------------------------------------------------------
  [1XCopyright[101X
  [33X[0;0YThis  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.[133X
  
  [33X[0;0YThis  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.[133X
  
  [33X[0;0YYou should have received a copy of the GNU General Public License along with
  this program. If not, see [7Xhttps://www.gnu.org/licenses/[107X[133X
  
  
  -------------------------------------------------------
  [1XAcknowledgements[101X
  [33X[0;0YTo  begin  with we are very grateful for all the feedback by users of former
  versions of [5XAlnuth[105X.[133X
  
  [33X[0;0YWe  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.[133X
  
  [33X[0;0YFor feedback on the development version, including a code patch, we are much
  obliged to Max Horn.[133X
  
  [33X[0;0YThe  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 [5XAlnuth[105X.[133X
  
  [33X[0;0YSupport for using OSCAR instead of PARI/GP was added by Claus Fieker and Max
  Horn.[133X
  
  
  -------------------------------------------------------
  
  
  [1XContents (Alnuth)[101X
  
  1 [33X[0;0YIntroduction[133X
  2 [33X[0;0YWorking with number fields[133X
    2.1 [33X[0;0YCreation of number fields[133X
      2.1-1 FieldByMatricesNC
      2.1-2 FieldByMatrixBasisNC
      2.1-3 FieldByPolynomialNC
    2.2 [33X[0;0YMethods for number fields[133X
      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 [33X[0;0YPresentations of multiplicative subgroups[133X
      2.3-1 PcpPresentationOfMultiplicativeSubgroup
      2.3-2 RelationLattice
    2.4 [33X[0;0YMethods to compute with subgroups of the unit group[133X
      2.4-1 RelationLatticeOfUnits
      2.4-2 IntersectionOfUnitSubgroups
    2.5 [33X[0;0YFactorisation of polynomials over a number field[133X
      2.5-1 FactorsPolynomialAlgExt
      2.5-2 FactorsPolynomialAlnuth
    2.6 [33X[0;0YExamples[133X
      2.6-1 ExampleMatField
  3 [33X[0;0YAn example application[133X
    3.1 [33X[0;0YNumber fields defined by matrices[133X
    3.2 [33X[0;0YNumber fields defined by a polynomial[133X
  4 [33X[0;0YInstallation[133X
    4.1 [33X[0;0YInstalling Alnuth[133X
    4.2 [33X[0;0YGetting PARI/GP[133X
    4.3 [33X[0;0YAdjust the path of the executable for GP[133X
      4.3-1 PariVersion
      4.3-2 SetAlnuthExternalExecutable
      4.3-3 SetAlnuthExternalExecutablePermanently
      4.3-4 RestoreAlnuthExternalExecutablePermanently
    4.4 [33X[0;0YLoading and testing the package[133X
  
  
  [32X
