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Algorithmic algebraic number theory. (English) Zbl 0685.12001
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 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 .

Reviewer: F.van der Linden

MSC:
12-02Research monographs (field theory)
12-04Machine computation, programs (field theory)
11RxxAlgebraic number theory: global fields
11R21Other number fields
11R27Units and factorization
11R23Iwasawa theory
11R32Galois theory for global fields
11H55Quadratic forms