Class Field Theory

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

From the reviews:

"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 … class field theory, illustrated with many well-chosen examples. … should be an obligatory reading for everybody interested in the modern development of algebraic number theory." (Wladyslaw Narkiewicz, Zentralblatt MATH, Vol. 1019, 2003)

"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 … . This book … gives much material in an elementary way, and is suitable for students,researchers and all who are fascinated by this theory." (L’Enseignement Mathematique, Vol. 49 (1-2), 2003)

"Each subject is treated very clearly from the theoretical side and explained by examples. The richness in examples is among the most attractive features of this book. … The book concludes with a very ample and well-organized bibliography. The writing is very clear and precise throughout. … This book gives an encompassing theoretical picture of large parts of class field theory. It is of particular interest to everybody interested … in this domain. … it is also a very enjoyable book." (Cornelius Greither, Mathematical Reviews, 2003 j)

• Faculty of Sciences, Laboratory of Mathematics and CNRS, University of Franche-Comté, Besançon Cedex, France

  Georges Gras

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