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Algebraic number theory. (English) Zbl 0744.11001
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: V.Ennola (Turku)

11-01Textbooks (number theory)
11RxxAlgebraic number theory: global fields
11R27Units and factorization
11R33Integral representations related to algebraic numbers
11R29Class numbers, class groups, discriminants
11D41Higher degree diophantine equations
11R18Cyclotomic extensions
11R42Zeta functions and $L$-functions of global number fields
11D25Cubic and quartic diophantine equations
11R11Quadratic extensions
11R16Cubic and quartic extensions
11R21Other number fields