Algorithmic algebraic number theory. (English) Zbl 0685.12001

Pohst, M.; Zassenhaus, Hans

Encyclopedia of Mathematics and its Applications, 30. Cambridge etc.: Cambridge University Press. xiv, 465 p. £50.00; $ 89.56 (1989).

This book gives an introduction to algebraic number theory. The authors concentrate on the algorithmic aspects of the theory. Many algorithms are given to compute properties of algebraic number fields and their subrings. The book deals with the following subjects: Galois theory, resolvents and discriminants, normal bases, geometry of numbers (lattice reduction), valuation theory, Newton polygon, units and computation of the class group.
The book finishes with a collection of tables. These tables involve permutation groups of degree \(\leq 12\), fundamental units and class groups of fields with degrees up to 7. The last table contains two computations of integral bases. The first one is for an 11th degree field, the second one for a 55th degree field. Both are given by a polynomial over \({\mathbb{Z}}\).

Reviewer: F. J. van der Linden (Eindhoven)

MSC:

11-02	Research exposition (monographs, survey articles) pertaining to number theory
11Yxx	Computational number theory
11Y40	Algebraic number theory computations
11Rxx	Algebraic number theory: global fields
11R21	Other number fields
11R27	Units and factorization
11R29	Class numbers, class groups, discriminants
11R32	Galois theory
11H55	Quadratic forms (reduction theory, extreme forms, etc.)

Keywords:

computational number theory; algorithms; lattice reduction; units; class group