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Local Fields

  • Textbook
  • © 1979

Overview

Authors:
  1. Jean-Pierre Serre

    1. Collège de France, Paris, France

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Part of the book series: Graduate Texts in Mathematics (GTM, volume 67)

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Table of contents (16 chapters)

  1. Front Matter

    Pages i-viii
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  2. Introduction

    1. Introduction

      • Jean-Pierre Serre
      Pages 1-2

  3. Local Fields (Basic Facts)

    1. Front Matter

      Pages 3-3
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    2. Discrete Valuation Rings and Dedekind Domains

      • Jean-Pierre Serre
      Pages 5-25

    3. Completion

      • Jean-Pierre Serre
      Pages 26-44

  4. Ramification

    1. Front Matter

      Pages 45-45
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    2. Discriminant and Different

      • Jean-Pierre Serre
      Pages 47-60

    3. Ramification Groups

      • Jean-Pierre Serre
      Pages 61-79

    4. The Norm

      • Jean-Pierre Serre
      Pages 80-96

    5. Artin Representation

      • Jean-Pierre Serre
      Pages 97-106

  5. Group Cohomology

    1. Front Matter

      Pages 107-107
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    2. Basic Facts

      • Jean-Pierre Serre
      Pages 109-126

    3. Cohomology of Finite Groups

      • Jean-Pierre Serre
      Pages 127-137

    4. Theorems of Tate and Nakayama

      • Jean-Pierre Serre
      Pages 138-149

    5. Galois Cohomology

      • Jean-Pierre Serre
      Pages 150-163

    6. Class Formations

      • Jean-Pierre Serre
      Pages 164-178

  6. Local Class Field Theory

    1. Front Matter

      Pages 179-179
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    2. Brauer Group of a Local Field

      • Jean-Pierre Serre
      Pages 181-187

    3. Local Class Field Theory

      • Jean-Pierre Serre
      Pages 188-203

    4. Local Symbols and Existence Theorem

      • Jean-Pierre Serre
      Pages 204-222
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Keywords

About this book

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.

Authors and Affiliations

  • Collège de France, Paris, France

    Jean-Pierre Serre

Bibliographic Information

  • Book Title: Local Fields

  • Authors: Jean-Pierre Serre

  • Series Title: Graduate Texts in Mathematics

  • DOI: https://doi.org/10.1007/978-1-4757-5673-9

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 1979

  • Hardcover ISBN: 978-0-387-90424-5Published: 19 January 1980

  • Softcover ISBN: 978-1-4757-5675-3Published: 31 May 2013

  • eBook ISBN: 978-1-4757-5673-9Published: 29 June 2013

  • Series ISSN: 0072-5285

  • Series E-ISSN: 2197-5612

  • 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: Algebra

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