# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 $\le 12$, 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 $ℤ$.

Reviewer: F.van der Linden

##### MSC:
 12-02 Research monographs (field theory) 12-04 Machine computation, programs (field theory) 11Rxx Algebraic number theory: global fields 11R21 Other number fields 11R27 Units and factorization 11R23 Iwasawa theory 11R32 Galois theory for global fields 11H55 Quadratic forms