## 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).

### 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