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.
