Local Fields

1. Front Matter

   Pages i-viii

   PDF

2. Introduction

   1. Introduction

      • Jean-Pierre Serre

      Pages 1-2

3. Local Fields (Basic Facts)

   1. Front Matter

      Pages 3-3

      PDF

   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

      PDF

   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

      PDF

   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

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

• Collège de France, Paris, France

  Jean-Pierre Serre

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