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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