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Graduate Texts in Mathematics

Local Fields

Authors: Serre, Jean-Pierre

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About this Textbook

The goal of this book is to present local class field theory from the cohomo­ logical point of view, following the method inaugurated by Hochschild and developed by Artin-Tate. This theory is about extensions-primarily abelian-of "local" (i.e., complete for a discrete valuation) fields with finite residue field. For example, such fields are obtained by completing an algebraic number field; that is one of the aspects of "localisation". The chapters are grouped in "parts". There are three preliminary parts: the first two on the general theory of local fields, the third on group coho­ mology. Local class field theory, strictly speaking, does not appear until the fourth part. Here is a more precise outline of the contents of these four parts: The first contains basic definitions and results on discrete valuation rings, Dedekind domains (which are their "globalisation") and the completion process. The prerequisite for this part is a knowledge of elementary notions of algebra and topology, which may be found for instance in Bourbaki. The second part is concerned with ramification phenomena (different, discriminant, ramification groups, Artin representation). Just as in the first part, no assumptions are made here about the residue fields. It is in this setting that the "norm" map is studied; I have expressed the results in terms of "additive polynomials" and of "multiplicative polynomials", since using the language of algebraic geometry would have led me too far astray.

Table of contents (16 chapters)

Table of contents (16 chapters)

Buy this book

eBook $54.99
price for USA in USD (gross)
  • ISBN 978-1-4757-5673-9
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $74.95
price for USA in USD
  • ISBN 978-0-387-90424-5
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $74.95
price for USA in USD
  • ISBN 978-1-4757-5675-3
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Local Fields
Authors
Translated by
Greenberg, M.J.
Series Title
Graduate Texts in Mathematics
Series Volume
67
Copyright
1979
Publisher
Springer-Verlag New York
Copyright Holder
Springer Science+Business Media New York
eBook ISBN
978-1-4757-5673-9
DOI
10.1007/978-1-4757-5673-9
Hardcover ISBN
978-0-387-90424-5
Softcover ISBN
978-1-4757-5675-3
Series ISSN
0072-5285
Edition Number
1
Number of Pages
VIII, 241
Number of Illustrations
62 b/w illustrations
Additional Information
Title of the original French edition: Corps locaux
Topics