A Gentle Course in Local Class Field Theory: Local Number Fields, Brauer Groups, Galois CohomologyThis book offers a self-contained exposition of local class field theory, serving as a second course on Galois theory. It opens with a discussion of several fundamental topics in algebra, such as profinite groups, p-adic fields, semisimple algebras and their modules, and homological algebra with the example of group cohomology. The book culminates with the description of the abelian extensions of local number fields, as well as the celebrated Kronecker-Weber theory, in both the local and global cases. The material will find use across disciplines, including number theory, representation theory, algebraic geometry, and algebraic topology. Written for beginning graduate students and advanced undergraduates, this book can be used in the classroom or for independent study. |
Contents
Kummer theory | 3 |
Local number fields | 23 |
Tools from topology | 54 |
The multiplicative structure of local number fields | 72 |
BRAUER GROUPS | 83 |
Central simple algebras | 107 |
Combinatorial constructions | 122 |
The Brauer group of a local number field | 146 |
Group cohomology | 187 |
Hilbert go | 204 |
Finer structure | 214 |
CLASS FIELD THEORY | 241 |
An introduction to number fields | 261 |
background material | 281 |
| 290 | |
GALOIS COHOMOLOGY | 157 |
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A Gentle Course in Local Class Field Theory: Local Number Fields, Brauer ... Pierre Guillot No preview available - 2018 |
Common terms and phrases
A-module abelian extension abelian group algebraic closure assume Br(F Brauer group called chain complex chapter class field theory cochain cocycle coefficients cohomology groups commutative compact complete contained Corollary cyclic group deduce define definition degree dimension direct sum element Enda Example exists extension K/F F-algebra fact field F finite group finite-dimensional follows G₁ Gal(K/F Galois extension Galois theory group G homomorphism homotopy ideal identity induced injective integer isomorphism kernel Kummer theory Lemma Let F Let G long exact sequence Mn(F morphism multiplication nonzero norm notation nth root number field p-adic fields prime number profinite group projective resolution Proof Proposition prove ramified reader ring root of unity semisimple short exact sequence simple modules skewfield splitting field subgroup submodule subset Suppose surjective theorem topological group topology trivial unique unramified valuation vector space write Z[G]-module Z/nZ