Class field theory: From theory to practice
Résumé
Global class field theory is a major achievement of algebraic number theory, based on the Artin 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, Grunwald--Wang's 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 puts emphasis on the invariant 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 recent computational one, gives much material in an elementary way, and is suitable for students, researchers, and all those 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. ----------------------------------------------------------------------------------------------------------------------------------------http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1019.11032&format=complete --------------------------------------------------------------------------------------------------------------------------------- FOR A COMPLETE VIEW OF MY PUBLICATIONS, PLEASE LOOK AT MY HOME PAGE:
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