<Chapter Label="Introduction">
<Heading>Introduction</Heading>

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 <Cite Key="Sta79"/>.
<P/>

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 <Cite Key="PARI2"/> respectively OSCAR <Cite Key="Oscar25"/>, <Cite Key="OSCAR"/>.
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;.
<P/>

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.
<P/>

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 <Cite Key="Poh93"/>,
<Cite Key="PZa89"/> and <Cite Key="Coh93"/>.
<P/>

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.
<P/>

</Chapter>

