Local Fields

Description

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. Front Matter....Pages i-viii Introduction....Pages 1-2 Front Matter....Pages 3-3 Discrete Valuation Rings and Dedekind Domains....Pages 5-25 Completion....Pages 26-44 Front Matter....Pages 45-45 Discriminant and Different....Pages 47-60 Ramification Groups....Pages 61-79 The Norm....Pages 80-96 Artin Representation....Pages 97-106 Front Matter....Pages 107-107 Basic Facts....Pages 109-126 Cohomology of Finite Groups....Pages 127-137 Theorems of Tate and Nakayama....Pages 138-149 Galois Cohomology....Pages 150-163 Class Formations....Pages 164-178 Front Matter....Pages 179-179 Brauer Group of a Local Field....Pages 181-187 Local Class Field Theory....Pages 188-203 Local Symbols and Existence Theorem....Pages 204-222 Ramification....Pages 223-231 Back Matter....Pages 232-245