Class Field Theory: From Theory to Practice

Description

Global class field theory is a major achievement of algebraic number theory, based on the functorial properties of the reciprocity map and the existence theorem. The author works out the consequences and the practical use of these results by giving detailed studies and illustrations of classical subjects (classes, idèles, ray class fields, symbols, reciprocity laws, Hasse's principles, the Grunwald-Wang theorem, Hilbert's towers,...). He also proves some new or less-known results (reflection theorem, structure of the abelian closure of a number field) and lays emphasis on the invariant (/cal T) p, of abelian p-ramification, which is related to important Galois cohomology properties and p-adic conjectures. This book, intermediary between the classical literature published in the sixties and the recent computational literature, gives much material in an elementary way, and is suitable for students, researchers, and all who are fascinated by this theory. In the corrected 2nd printing 2005, the author improves some mathematical and bibliographical details and adds a few pages about rank computations for the general reflection theorem; then he gives an arithmetical interpretation for usual class groups, and applies this to the Spiegelungssatz for quadratic fields and for the p-th cyclotomic field regarding the Kummer--Vandiver conjecture in a probabilistic point of view. Front Matter....Pages i-xiii Introduction to Global Class Field Theory....Pages 1-6 Basic Tools and Notations....Pages 7-63 Reciprocity Maps Existence Theorems....Pages 65-219 Abelian Extensions with Restricted Ramification — Abelian Closure....Pages 221-360 Invariant Class Groups in p -Ramification Genus Theory....Pages 361-405 Cyclic Extensions with Prescribed Ramification....Pages 407-439 Erratum....Pages 495-512 Back Matter....Pages 441-494