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# 1 Introduction

### Sections

1. Acknowledgements
2. License

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 Sta79.

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 System PARI/GP PARI2. Hence this package contains some GAP functions and an interface to some functions in the computer algebra system PARI/GP. Therefore one has to have PARI/GP installed to use the full functionality of Alnuth.

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

The main methods included in Alnuth are: creating a number field, computing its maximal order (using PARI/GP), computing its unit group (using PARI/GP) and a presentation of this unit group, computing the elements of a given norm of the number field (using PARI/GP), determining a presentation for a finitely generated multiplicative subgroup (using PARI/GP), and factoring polynomials defined over number fields (using PARI/GP). For background on algorithms for number fields we refer to Poh93, PZa89 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, are described.

## 1.1 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.

## 1.2 License

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/

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