EAN: 9781475756753

LOCAL FIELDS - JEAN-PIERRE SERRE KARTONIERT (TB)

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.

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