×

Galois theory of \(p\)-extensions. Transl. from the German by F. Lemmermeyer. (English) Zbl 1023.11002

Springer Monographs in Mathematics. Berlin: Springer. xiii, 190 p. EUR 69.95/net; sFr 116.50; £49.00; $ 79.95 (2002).
This is an English translation (an excellent job by F. Lemmermeyer) of the German edition which appeared in 1970 (see the review in Zbl 0216.04704) (the Russian translation is available since 1973). Although there is no need in detailed description of this by now classical book, here is a brief account of its content. The first half of the monograph is devoted to the theory of pro-\(p\)-groups and their cohomology. In spite of the fact that many new textbooks on this topic are available nowadays, Koch’s clear and precise exposition may still serve as an excellent guide to any newcomer. The main object of the second part of the book is the Galois group \(G_S\) of the maximal \(p\)-extension \(k_S\) of a global field \(k\) unramified outside some given set of primes \(S\) of \(k\). The goal is to compute the generators and relations of \(G_S\), its cohomological dimension and to apply these results to the construction of infinite class field towers.
A recent encyclopedic monograph by J. Neukirch, A. Schmidt and K. Wingberg [Cohomology of number fields, Springer-Verlag, Berlin (2000; Zbl 0948.11001)] treats many similar topics. However, as the author notes in the preface for the English translation justifying its appearance, there the focus is made on the case of maximal \(p\)-extensions unramified outside of a set \(S\) of primes containing the primes dividing \(p\); in the book under review the author concentrates on extensions in which primes dividing \(p\) may be unramified. In addition, some important results (like the improved version of the Golod-Shafarevich inequality) are discussed in the above cited monograph only marginally.
One should note the postscript to the book written by the author and the translator. It contains some interesting historic material on the subject (on the “prehistoric” period, between Dedekind and Shafarevich, as well as on the “Golod-Shafarevich” period where the author gives an account of his early collaboration with Shafarevich). It also contains a survey on the development of the field in the last 30 years. In particular, there is an overview of results on the structure of \(G_S\) (mainly due to L. V. Kuzmin); various generalizations of the criterion of Golod-Shafarevich for infinite class field towers; group-theoretical investigations on the possible Galois groups of finite class field towers; applications of infinite class field towers to discriminant bounds; structure of the first quotients in the lower central series of \(G_S\); the Fontaine-Mazur conjecture and related problems. Also, a brief overview of recent textbooks on the subject has been included.

MSC:

11-02 Research exposition (monographs, survey articles) pertaining to number theory
11R34 Galois cohomology
11R32 Galois theory
PDFBibTeX XMLCite