''Corps Locaux'' by
Jean-Pierre Serre
Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ...
, originally published in 1962 and translated into English as ''Local Fields'' by
Marvin Jay Greenberg in 1979, is a seminal graduate-level
algebraic number theory text covering
local fields,
ramification,
group cohomology
In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomolog ...
, and
local class field theory In mathematics, local class field theory, introduced by Helmut Hasse, is the study of abelian extensions of local fields; here, "local field" means a field which is complete with respect to an absolute value or a discrete valuation with a finite r ...
. The book's end goal is to present local class field theory from the cohomological point of view. This theory concerns extensions of "local" (i.e., complete for a discrete valuation) fields with finite residue field.
Contents
#''Part I, Local Fields (Basic Facts)'': Discrete valuation rings, Dedekind domains, and Completion.
#''Part II, Ramification'': Discriminant & Different, Ramification Groups, The Norm, and Artin Representation.
#''Part III, Group Cohomology'': Abelian & Nonabelian Cohomology, Cohomology of Finite Groups, Theorems of Tate and Nakayama, Galois Cohomology, Class Formations, and Computation of Cup Products.
#''Part IV, Local Class Field Theory'': Brauer Group of a Local Field, Local Class Field Theory, Local Symbols and Existence Theorem, and Ramification.
References
*
Algebraic number theory
Class field theory
Mathematics books
{{mathematics-lit-stub