Springer Monographs in Mathematics

Class Field Theory

From Theory to Practice

Authors: Gras, Georges

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eBook 79,72 €
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  • ISBN 978-3-662-11323-3
  • Digitally watermarked, DRM-free
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Hardcover 98,79 €
price for Spain (gross)
  • ISBN 978-3-540-44133-5
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Softcover 98,79 €
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  • ISBN 978-3-642-07908-5
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  • The final prices may differ from the prices shown due to specifics of VAT rules
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.

Reviews

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)


Table of contents (7 chapters)

  • Introduction to Global Class Field Theory

    Gras, Georges

    Pages 1-6

    Preview Buy Chapter 30,19 €
  • Basic Tools and Notations

    Gras, Georges

    Pages 7-63

    Preview Buy Chapter 30,19 €
  • Reciprocity Maps Existence Theorems

    Gras, Georges

    Pages 65-219

    Preview Buy Chapter 30,19 €
  • Abelian Extensions with Restricted Ramification — Abelian Closure

    Gras, Georges

    Pages 221-360

    Preview Buy Chapter 30,19 €
  • Invariant Class Groups in p-Ramification Genus Theory

    Gras, Georges

    Pages 361-405

    Preview Buy Chapter 30,19 €

Buy this book

eBook 79,72 €
price for Spain (gross)
  • ISBN 978-3-662-11323-3
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover 98,79 €
price for Spain (gross)
  • ISBN 978-3-540-44133-5
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
  • The final prices may differ from the prices shown due to specifics of VAT rules
Softcover 98,79 €
price for Spain (gross)
  • ISBN 978-3-642-07908-5
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
  • The final prices may differ from the prices shown due to specifics of VAT rules
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Bibliographic Information

Bibliographic Information
Book Title
Class Field Theory
Book Subtitle
From Theory to Practice
Authors
Translated by
Cohen, H.
Series Title
Springer Monographs in Mathematics
Copyright
2003
Publisher
Springer-Verlag Berlin Heidelberg
Copyright Holder
Springer-Verlag Berlin Heidelberg
eBook ISBN
978-3-662-11323-3
DOI
10.1007/978-3-662-11323-3
Hardcover ISBN
978-3-540-44133-5
Softcover ISBN
978-3-642-07908-5
Series ISSN
1439-7382
Edition Number
1
Number of Pages
XIII, 491
Topics