Sophie: gap-system-packages-4.4.12-5mdv2010.0 i586

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<html><head><title>[ALNUTH] 1 Introduction</title></head>
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<h1>1 Introduction</h1><p>
<p>
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 <a href="biblio.htm#Sta79"><cite>Sta79</cite></a>.
<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 
System KANT <a href="biblio.htm#Kant"><cite>Kant</cite></a>. 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.
<p>
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. 
<p>
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 <a href="biblio.htm#Poh93"><cite>Poh93</cite></a>, <a href="biblio.htm#PZa89"><cite>PZa89</cite></a> and
<a href="biblio.htm#Coh93"><cite>Coh93</cite></a>.
<p>
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. 
<p>
<p>
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<P>
<address>ALNUTH manual<br>November 2006
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