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Notes on class field theory
Kiran S. Kedlaya
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Contents
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Front Matter
Colophon
Preface
1
Trailer: Abelian extensions of the rationals
The Kronecker-Weber theorem
Kummer theory
The local Kronecker-Weber theorem
2
The statements of class field theory
The Hilbert class field
Generalized ideal class groups and the Artin reciprocity law
The principal ideal theorem
Zeta functions and the Chebotaryov density theorem
3
Cohomology of groups
Cohomology of finite groups I: abstract nonsense
Cohomology of finite groups II: concrete nonsense
Homology and Tate groups
Cohomology of cyclic groups
Profinite groups and infinite Galois theory
4
Local class field theory
Overview of local class field theory
Cohomology of local fields: some computations
Local class field theory via Tate's theorem
Ramification filtrations and local reciprocity
Making the reciprocity map explicit
5
Abstract class field theory
The setup of abstract class field theory
The abstract reciprocity map
The theorems of abstract class field theory
A look ahead
6
The adelic formulation
Adèles
Idèles and class groups
Adèles and idèles in field extensions
The adelic reciprocity law and Artin reciprocity
Adelic reciprocity: what remains to be done
Adelic Fourier analysis after Tate
7
The main results
Cohomology of the idèles I: the “First Inequality”
Cohomology of the idèles II: the “Second Inequality”
An “abstract” reciprocity map
The existence theorem
Local-global compatibility
Brauer groups and the reciprocity map
Back Matter
A
Parting thoughts
Bibliography
Authored in PreTeXt
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Front Matter
1
Trailer: Abelian extensions of the rationals
2
The statements of class field theory
3
Cohomology of groups
4
Local class field theory
5
Abstract class field theory
6
The adelic formulation
7
The main results
Back Matter
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