Springer Monographs in Mathematics. Berlin: Springer. xiii, 491 p. EUR 79.95/net; sFr 133.00; £ 56.00; $ 79.95 (2003).
The author writes in the preface that the aim of this book is “to help in the practical use and understanding of the principles of global class field theory for number fields, without any attempt to give proofs of the foundations …”. He succeeded in his task admirably. The book brings a huge amount of information on the main results and applications of class field theory, illustrated with many well-chosen examples. It consists of five chapters:
The first introduces the main notions -- (absolute values, ideles, class groups, …) and presents their properties, and the second chapter surveys (mostly without proofs) the fundamental theorems of class field theory, both local and global, the latter presented in both its versions -- idelic and classical, based on congruence groups. One also finds here a study of maximal Abelian extensions and a discussion of questions connected with Leopoldt’s “Spiegelungssatz”, principal ideal theorem, the capitulation problem, class field towers, Hasse’s norm theorem and symbols. This is illustrated by several examples, which include a sextic field, found by Maire and Hajir, having class-number one and an infinite class field tower.
The next chapter concerns the properties and structure of Abelian extensions with given ramification, both finite and infinite. We find here among other things a discussion on the logarithmic class group and on Leopoldt’s conjecture on the $p$-adic unit rank as well as its generalization due to Jaulent and Roy. Also the Grunwald-Wang theorem finds its place here.
The fourth chapter, after a study of fields satisfying Leopoldt’s conjecture for a prime $p$ (the $p$-rational fields), brings a thorough treatment of the genus theory. The final chapter deals with cyclic extensions with a prescribed ramification.
The book contains a huge amount of material, most of which appears for the first time in a book, so it is not an easy reading, but many exercises, often accompanied by hints of solutions, and several examples and computations make this task lighter. This book should be an obligatory reading for everybody interested in the modern development of algebraic number theory.