Class Field Theory - J.S. Milne   Top
Course Notes
Group Theory
Fields and Galois Theory
Algebraic Geometry
Algebraic Number Theory
Modular Functions and Modular Forms
Elliptic Curves -- see books.
Abelian Varieties
Lectures on Etale Cohomology
Class Field Theory
Complex Multiplication
Basic Theory of Affine Group Schemes
Lie Algebras, Algebraic Groups, and Lie Groups
Reductive Groups

pdf file for the current version (4.02)

This is a course on Class Field Theory, roughly along the lines of Artin and Tate and of the articles of Serre and Tate in Cassels-Fröhlich, except that the notes are more detailed and cover more.


  1. Local Class Field Theory: Lubin-Tate Extensions
  2. Cohomology of Groups.
  3. Local Class Field Theory Continued.
  4. Brauer Groups
  5. Global Class Field Theory: Statements
  6. L-series and the Density of Primes
  7. Global Class Field Theory: Proofs
  8. Complements (Power reciprocity laws; quadratic forms; etc.)


The algebra usually covered in first-year graduate courses and a course in algebraic number theory (for example, my online notes).


v2.01; first version on the web.
v3.10; May 6, 1997; substantially revised and expanded from v2.01; index; 222 pages.
v4.00; March 2, 2008; corrected, revised, and expanded; 287 pages.
v4.01; May 30, 2011; corrected; 287 pages.
v4.02; March 23, 2013; corrected; 289 pages.

pdf (old version 3.10)