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.
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圖書目錄
Introduction Leitfaden Part One LOCAL FIELDS (BASIC FACTS) 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. De&kind 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. Strupture of complete discrecte valuation rings I:Equal characteristic case 5. Structure of complete discrete valuation rings II:Unequal characteristic case 6. Witt vectors …… Chapter IV Ramification Groups Chapter V The Norm Chapter VI Artin Representation Part Three GROUP COHOMOLOGY Chapter VII Basic Facts Appendix Non-abelian Cohomology Chapter VIII Cohomology of Finite Groups Chapter IX Theorems of Tate and Nakayama Chapter X Galois Cohomology Chapter Xi Class Formations Part Four LOCAL CALSS FIELD THEORY Chapter XII Brauer group of a local field Chapter XIII Local class field theory ChapterXIV Local symbols and Existence Theorem Appendix The global case Chater XV Ramification Bibliography Supplementary Bibliography for the English Edition
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