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# Alnuth

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

4.0.2

21 June 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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