Milne J.S. Class Field Theory

Web Publication (Version 4.02; March 23, 2013). — [v2.01 (August 21, 1996). First version on the web.], 2013. — (281+viii) p., eBook, English (Interactive menu). (Free Access).Class field theory describes the abelian extensions of a local or global field in terms of the arithmetic of the field itself. These notes contain an exposition of abelian class field theory using the algebraic/cohomological approach of Chevalley and Artin and Tate. The explicit approach of Lubin and Tate in the local case and the analytic approach in the global case are also explained. The original version of the notes was distributed during the teaching of an advanced graduate course.Local Class Field Theory: Lubin-Tate Theory.
Statements of the Main Theorems.
Lubin-Tate Formal Group Laws.
Construction of the extension K of K.
The Local Kronecker-Weber Theorem.
A Appendix: Infinite Galois Theory and Inverse Limits.
The Cohomology of Groups.
Cohomology.
Homology.
The Tate groups.
The Cohomology of Profinite Groups.
A Appendix: Some Homological Algebra.
Local Class Field Theory: Cohomology.
The Cohomology of Unramified Extensions.
The Cohomology of Ramified Extensions.
The Local Artin Map.
The Hilbert symbol.
The Existence Theorem.
Brauer Groups.
Simple Algebras; Semisimple Modules.
Definition of the Brauer Group.
The Brauer Group and Cohomology.
The Brauer Groups of Special Fields.
Complements.
Global Class Field Theory: Statements.
Ray Class Groups.
L-series.
The Main Theorems in Terms of Ideals.
Ideles.
The Main Theorms in Terms of Ideles.
L-Series and the Density of Primes.
Dirichlet series and Euler products.
Convergence Results.
Density of the Prime Ideals Splitting in an Extension.
Density of the Prime Ideals in an Arithmetic Progression.
Global Class Field Theory: Proofs.
Outline.
The Cohomology of the Ideles.
The Cohomology of the Units.
Cohomology of the Idele Classes I: the First Inequality.
Cohomology of the Idele Classes II: The Second Inequality.
The Algebraic Proof of the Second Inequality.
Application to the Brauer Group.
Completion of the Proof of the Reciprocity Law.
The Existence Theorem.
A Appendix: Kummer theory.
Complements.
When are local nth powers global nth powers?
The Grunwald-Wang Theorem.
The local-global principle for norms and quadratic forms.
The Fundamental Exact Sequence and the Fundamental Class.
Higher Reciprocity Laws.
The Classification of Quadratic Forms over a Number Field.
Density Theorems.
Function Fields.
Cohomology of Number Fields.
More on L-series.
A Exercises.
B Solutions to Exercises.
C Sources for the history of class field theory.