Problems in the arithmetic of fields and Galois theory. (Problèmes d’arithmétique des corps et de théorie de Galois.) (French) Zbl 0912.12001

Deschamps, Bruno

Collection Méthodes. Paris: Hermann. 248 p. (1998).

The aim of this book is two-fold: first, to present some well-known facts in the arithmetic of fields and Galois theory, including: polynomials and cyclotomic fields, finite fields, the quadratic reciprocity law, the fundamental theorem of algebra, the four squares theorem, etc., and second, to discuss more special topics, including skew fields, inverse Galois theory, differential fields, ordered fields, etc.
This is a very nice book, very carefully written, with complete proofs of the results presented and complete solutions of the exercises, covering in few pages a large spectrum of an interesting topic in field theory. The book is recommended to graduates students, and to anyone wishing to enlarge his/her standard knowledge in this beautiful area of algebra and number theory, as well.

Reviewer: Toma Albu (Columbus)

MSC:

12-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to field theory
12E20	Finite fields (field-theoretic aspects)
12F10	Separable extensions, Galois theory
00A07	Problem books
12E15	Skew fields, division rings
12J15	Ordered fields
12F12	Inverse Galois theory
12H05	Differential algebra

Keywords:

arithmetic of fields; Galois theory; polynomials; cyclotomic fields; finite fields; quadratic reciprocity law; fundamental theorem of algebra; skew fields; inverse Galois theory; differential fields; ordered fields