This book is a first step in a new direction: to modify existing theory from a constructive point of view and to stimulate the readers to make their own computational experiments. We are thoroughly convinced that their observations will help to build a new basis from which to venture into new theory on algebraic numbers. History shows that in the long run, number theory always followed the cyclic movement from theory to construction to experiment to conjecture to theory.

Consequently, this book is addressed to all lovers of number theory. On the one hand, it gives a comprehensive introduction to (constructive) algebraic number theory and is therefore especially suited as a textbook for a course on that subject. On the other hand, many parts go far beyond an introduction and make the user familiar with recent research in the field. For experimental number theoreticians we developed new methods and obtained new results (e.g., in the tables at the end of the book) of great importance for them. Both computer scientists interested in higher arithmetic and in the basic makeup of digital computers, and amateurs and teachers liking algebraic number theory will find the book of value.

Many parts of the book have been tested in courses independently by both authors. However, the outcome is not presented in the form of lectures, but, rather, in the form of developed methods and problems to be solved. Algorithms occur frequently throughout the presentation.

Introduction

Since the first printing of this book in 1989 algorithmic algebraic number theory has attracted rapidly increasing interest. This is documented, for example, by a regular meeting, ANTS (algebraic number theory symposium), every two years whose proceedings give a good survey about ongoing research. Also there are several computer algebra packages concentrating on number theoretical computations. At present the most prominent ones, which are available for free, are KANT, PARI and SIMATH. KANT comes with a data base for algebraic number fields, already containing more than a million fields of small degree. KANT is developed by the research group of the author at Berlin and will be presented in some detail in this chapter. We note that almost all of KANT and PARI is also contained in the MAGMA system.

In the sequel we shortly discuss the improvements which were obtained for the computation of the important invariants of algebraic number fields. On the other hand, in computational algebraic number theory the interest has gradually turned from absolute extensions to relative extensions of number fields and we will sketch the important developments in that area. If subfields exist, the information about the invariants of those subfields can be lifted and used in the field under consideration. This relative point of view permits computations in fields of much larger degrees and has important applications, for example to class field computations.