Class Field Theory
1
Introduction
2
Group Cohomology
▶
2.1
Introduction
2.2
Group cohomology and homology
2.3
Tate Cohomology
2.4
Coinduction and induction
2.5
Dimension-shifting
▶
2.5.1
Shifting \(\mathrm{up}\)
2.5.2
Shifting \(\mathrm{down}\)
2.6
The inflation-restriction sequence
2.7
Corestriction
2.8
Periodicity for finite cyclic groups
2.9
Herbrand quotients
2.10
The Triviality Criterion
2.11
The splitting module of a 2-cocycle
2.12
The Reciprocity Isomorphism
2.13
Compatibility in towers
3
Local Class Field Theory
▶
3.1
Notation and Preliminary results
3.2
The Herbrand quotient of \(l^{\times }\)
3.3
An upper bound for \(H^2(l/k,l^\times )\)
3.4
Fundamental classes in unramified extensions
3.5
Construction of fundamental classes
3.6
Some consequences of the reciprocity isomorphism
4
Global Class Field Theory
▶
4.1
Choice of \(S\)
4.2
The Herbrand quotient of the \(S\)-ideles
4.3
The Herbrand quotient of the \(S\)-units
4.4
Dirichlet Density
4.5
Some \(L\)-functions
4.6
The first inequality
5
References
Dependency graph
1 Introduction