Algebraic number theory. (English) Zbl 0744.11001

Fröhlich, A.; Taylor, M. J.

Cambridge Studies in Advanced Mathematics. 27. Cambridge (UK): Cambridge University Press. xiv, 355 p. (1990).

This is an excellent introduction to the subject written by distinguished scholars. It is more ambitious than an average textbook containing material such as: (1) a thorough treatment of module theory over Dedekind rings – a topic obviously close to hearts of the authors; (2) properties of differents and discriminants; (3) a short introduction to elliptic curves meant to encourage the reader to learn more; (4) Brauer relations between Dedekind zeta-functions. There are 93 exercises. Throughout the text great stress is laid on worked concrete numerical examples. As prerequisites the authors assume familiarity with elementary topology, Galois theory, and basic module theory including tensor products.
The chapter headings are: I. Algebraic foundations, II. Dedekind domains (valuations, completions and module theory), III. Extensions (decomposition, ramification, discriminants and differents), IV. Class-groups and units, V. Fields of low degree (concrete applications of general results to fields of degree six or less), VI. Cyclotomic fields (including Gauss sums and elliptic curves), VII. Diophantine equations (Fermat’s last theorem, quadratic forms, cubic equations), VIII. L-functions (including the Dedekind zeta-function, class number formulae and Brauer relations).

Reviewer: Veikko Ennola (Turku)

MSC:

11-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
11Rxx	Algebraic number theory: global fields
11R27	Units and factorization
11R33	Integral representations related to algebraic numbers; Galois module structure of rings of integers
11R29	Class numbers, class groups, discriminants
11D41	Higher degree equations; Fermat’s equation
11R18	Cyclotomic extensions
11R42	Zeta functions and \(L\)-functions of number fields
11D25	Cubic and quartic Diophantine equations
11R11	Quadratic extensions
11R16	Cubic and quartic extensions
11R21	Other number fields

Keywords:

module theory over Dedekind rings; differents; discriminants; introduction to elliptic curves; Brauer relations between Dedekind zeta- functions; Dedekind domains; ramification; class-groups; units; cyclotomic fields; Fermat’s last theorem; class number formulae