Galois Theory of p-Extensions

Authors: Helmut Koch



First published in German in 1970 and translated into Russian in 1973, this classic now becomes available in English. After introducing the theory of pro-p groups and their cohomology, it discusses presentations of the Galois groups G S of maximal p-extensions of number fields that are unramified outside a given set S of primes. It computes generators and relations as well as the cohomological dimension of some G S, and gives applications to infinite class field towers.The book demonstrates that the cohomology of groups is very useful for studying Galois theory of number fields; at the same time, it offers a down to earth introduction to the cohomological method. In a "Postscript" Helmut Koch and Franz Lemmermeyer give a survey on the development of the field in the last 30 years. Also, a list of additional, recent references has been included.

Table of contents (14 chapters)

Front Matter

Introduction

Profinite Groups

Galois Theory of Infinite Algebraic Extensions

Cohomology of Profinite Groups

Free pro-p Groups

Cohomological Dimension

Presentation of pro-p Groups

Group Algebras of pro-p Groups

Results from Algebraic Number Theory

The Maximal p-Extension

Local Fields of Finite Type

Global Fields of Finite Type

On p-Class Groups and p-Class Field Towers

The Cohomological Dimension of Gs

Back Matter