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