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Local Fields (Hardcover, 1st ed. 1979. Corr. 2nd printing 1995)
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Local Fields (Hardcover, 1st ed. 1979. Corr. 2nd printing 1995)
Series: Graduate Texts in Mathematics, 67
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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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