Pohst, M.; Zassenhaus, Hans 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 \(\leq 12\), 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) Cited in 8 ReviewsCited in 123 Documents 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 PDF BibTeX XML Cite \textit{M. Pohst} and \textit{H. Zassenhaus}, Algorithmic algebraic number theory. Cambridge etc.: Cambridge University Press (1989; Zbl 0685.12001) OpenURL Online Encyclopedia of Integer Sequences: Class number of real quadratic field Q(sqrt f), where f is the n-th squarefree number A005117(n). Squarefree numbers: numbers that are not divisible by a square greater than 1. Table of orders of primitive permutation groups by degree. Table of orders of transitive permutation groups by degree. Value of x corresponding to the minimal solution of the Pell equation x^2+d*y^2, as d runs through the squarefree numbers. y values corresponding to the x values in A023677. Discriminants of complex cubic fields (negated). Discriminants of totally real quartic fields. Discriminants of quartic fields with 2 complex conjugates (negated). Discriminants of totally complex quartic fields. Discriminants of totally real quintic fields. Discriminants of quintic fields with 2 complex conjugates. Discriminants of quintic fields with 4 complex conjugates. Discriminants of totally real sextic fields. Discriminants of totally complex sextic fields. Number of factors when x^3-x-1 is factorized mod the n-th prime.