The goal of this book is to present local class field theory from the cohomological 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 cohomology. Local class field theory, strictly speaking, does not appear until the fourth part.

Introduction

Leitfaden

Part One

LOCAL FIELDS (BASIC FACTS)

Chapter I

Discrete Valuation Rings and Dedekind Domains

§1. Definition of Discrete Valuation Ring

§2. Characterisations of Discrete Valuation Rings

§3. Dedekind Domains

§4. Extensions

§5. The Norm and Inclusion Homomorphisms

§6. Example: Simple Extensions

§7. Galois Extensions

§8. Frobenius Substitution

Chapter II

Completion

§1. Absolute Values and the Topology Defined by a Discrete Valuation

§2. Extensions of a Complete Field

§3. Extension and Completion

§4. Structure of Complete Discrete Valuation Rings I: Equal Characteristic Case

§5. Structure of Complete Discrete Valuation Rings II: Unequal Characteristic Case

§6. Witt Vectors

Part Two

RAMIFICATION

Chapter III

Discriminant and Different

§1. Lattices

§2. Discriminant of a Lattice with Respect to a Bilinear Form

§3. Discriminant and Different of a Separable Extension

§4. Elementary Properties of the Different and Discriminant

§5. Unramified Extensions

§6. Computation of Different and Discriminant

§7. A Differential Characterisation of the Different

Chapter IV

Ramification Groups

§1. Definition of the Ramification Groups; First Properties

§2. The Quotients Gi/Gi+1, i≥0

§3. The Functions φ and ψ; Herbrand's Theorem

§4. Example: Cyclotomic Extensions of the Field Qp

Chapter V

The Norm

§1. Lemmas

§2. The Unramified Case

§3. The Cyclic of Prime Order Totally Ramified Case

§4. Extension of the Residue Field in a Totally Ramified Extension

§5. Multiplicative Polynomials and Additive Polynomials

§6. The Galois Totally Ramified Case

§7. Application: Proof of the Hasse-Arf Theorem

Chapter VI

Artin Representation

§1. Representations and Characters

§2. Artin Representation

§3. Globalisation

§4. Artin Representation and Homology (for Algebraic Curves)

Part Three

GROUP COHOMOLOGY

Chapter VII

Basic Facts

§1. G-Modules

§2. Cohomology

§3. Computing the Cohomology via Cochains

§4. Homology

§5. Change of Group

§6. An Exact Sequence

§7. Subgroups of Finite Index

§8. Transfer

Appendix

Non-abelian Cohomology

Chapter VIII

Cohomology of Finite Groups

§1. The Tate Cohomology Groups

§2. Restriction and Corestri&ion

§3. Cup Products

§4. Cohomology of Finite Cyclic Groups. Herbrand Quotient

§5. Herbrand Quotient in the Cyclic of Prime Order Case

Chapter IX

Theorems of Tate and Nakayama

§1. p-Groups

§2. Sylow Subgroups

§3. Induced Modules; Cohomologically Trivial Modules

§4. Cohomology of a p-Group

§5. Cohomology of a Finite Group

§6. Dual Results

§7. Comparison Theorem

§8. The Theorem of Tate and Nakayama

Chapter X

Galois Cohomology

§1. First Examples

§2. Several Examples of "Descent"

§3. Infinite Galois Extensions

§4. The Brauer Group

§5. Comparison with the Classical Definition of the Brauer Group

§6. Geometric Interpretation of the Brauer Group: Severi-Brauer Varieties

§7. Examples of Brauer Groups

Chapter XI

Class Formations

§1. The Notion of Formation

§2. Class Formations

§3. Fundamental Classes and Reciprocity Isomorphism

§4. Abelian Extensions and Norm Groups

§5. The Existence Theorem

Appendix

Computations of Cup Products

Part Four

LOCAL CLASS FIELD THEORY

Chapter XII

Brauer Group of a Local Field

§1. Existence of an Unramified Splitting Field

§2. Existence of an Unramified Splitting Field (Direct Prool)

§3. Determination of the Brauer Group

Chapter XIII

Local Class Field Theory

§1. The Group Z and Its Cohomology

§2. Quasi-Finite Fields

§3. The Brauer Group

§4. Class Formation

§5. Dwork's Theorem

Chapter XIV

Local Symbols and Existence Theorem

§1. General Definition of Local Symbols

§2. The Symbol (a, b)

§3. Computation of the Symbol (a, b)v in the Tamely Ramified Case

§4. Computation of the Symbol (a, b)v for the Field Qp(n=2)

§5. The symbols [a, b)

§6. The Existence Theorem

§7. Example: The Maximal Abelian Extension of Qp

Appendix

The Global Case (Statement of Results)

Chapter XV

Ramification

§1. Kernel and Cokernel of an Additive (resp. Multiplicative) Polynomial

§2. The Norm Groups

§3. Explicit Computations

Bibliography

Supplementary Bibliography for the English Edition

Index