Local Fields

  • Jean-Pierre Serre

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

Table of contents

  1. Front Matter
    Pages i-viii
  2. Introduction

    1. Jean-Pierre Serre
      Pages 1-2
  3. Local Fields (Basic Facts)

    1. Front Matter
      Pages 3-3
    2. Jean-Pierre Serre
      Pages 5-25
    3. Jean-Pierre Serre
      Pages 26-44
  4. Ramification

    1. Front Matter
      Pages 45-45
    2. Jean-Pierre Serre
      Pages 47-60
    3. Jean-Pierre Serre
      Pages 61-79
    4. Jean-Pierre Serre
      Pages 80-96
    5. Jean-Pierre Serre
      Pages 97-106
  5. Group Cohomology

    1. Front Matter
      Pages 107-107
    2. Jean-Pierre Serre
      Pages 109-126
    3. Jean-Pierre Serre
      Pages 127-137
    4. Jean-Pierre Serre
      Pages 138-149
    5. Jean-Pierre Serre
      Pages 150-163
    6. Jean-Pierre Serre
      Pages 164-178
  6. Local Class Field Theory

    1. Front Matter
      Pages 179-179
    2. Jean-Pierre Serre
      Pages 181-187
    3. Jean-Pierre Serre
      Pages 188-203
    4. Jean-Pierre Serre
      Pages 204-222
    5. Jean-Pierre Serre
      Pages 223-231
  7. Back Matter
    Pages 232-245

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.


Fields Lokaler Körper algebra algebraic geometry algebraic number field arithmetic cohomology field finite group polynomial

Authors and affiliations

  • Jean-Pierre Serre
    • 1
  1. 1.Collège de FranceParisFrance

Bibliographic information

  • DOI
  • Copyright Information Springer Science+Business Media New York 1979
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4757-5675-3
  • Online ISBN 978-1-4757-5673-9
  • Series Print ISSN 0072-5285
  • Series Online ISSN 2197-5612
  • Buy this book on publisher's site