[ALNUTH] 1 Introduction

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# 1 Introduction

A number field is a finite extension of the field of rational numbers. This package 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 KANT Kant. Hence this package contains some Gap functions and an interface to some functions in the computer algebra system KANT. Therefore one has to have installed KANT to use the full functionality of this package. Furthermore the interface only runs with the Linux version of Gap.

We note that not all available functions of KANT are linked to Gap and the KANT system provides much more methods for computations in number fields.

The main methods included in this package are: creating a number field, computing its maximal order (using KANT), computing its unit group (using KANT) and a presentation of this unit group, computing the elements of a given norm of the number field (using KANT) and determining a presentation for a finitely generated multiplicative subgroup (using KANT). For background on algorithms for number fields we refer to Poh93, PZa89 and Coh93.

The functions provided by this package are introduced in the following chapter. Then an example application is outlined. In the final chapter of this manual the installation of the package is described. We note that the computer algebra system KANT itself is not included in the package.

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