Class Field Theory

From Theory to Practice

  • Georges Gras

Part of the Springer Monographs in Mathematics book series (SMM)

Table of contents

  1. Front Matter
    Pages i-xiii
  2. Georges Gras
    Pages 7-63
  3. Georges Gras
    Pages 65-219
  4. Georges Gras
    Pages 495-512
  5. Back Matter
    Pages 441-494

About this book


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.


Abelian closure Class field theory algebra idele groups number fields number theory reciprocity laws

Authors and affiliations

  • Georges Gras
    • 1
  1. 1.Faculty of Sciences, Laboratory of Mathematics and CNRSUniversity of Franche-ComtéBesançon CedexFrance

Bibliographic information

  • DOI
  • Copyright Information Springer-Verlag Berlin Heidelberg 2003
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-07908-5
  • Online ISBN 978-3-662-11323-3
  • Series Print ISSN 1439-7382
  • Buy this book on publisher's site
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