Math 249

Over two quarters, this course will focus on the class field theory, including the construction of the Weil group and the theories of Hecke and Artin L-functions. We will begin with class field theory. The course should be quite different from Rubin's course last year, however.

Announcements

During Winter Quarter, the class will meet MWF 9-10 in 381T.
Syllabus for the Winter Quarter.

Course materials

Introductory lecture. Revised Thursday Sept. 30.
Quadratic Extensions of Local and Global Fields. Revised December 1, 2004. (Proof of Theorem 9 completed.)
Here is the same notes as a PDF file.

Other resources

Hochschild's review of Weil's 1951 paper on class field theory.
Peter Roquette's homepage contains many historical papers on the origins of modern algebraic number theory. For example, the paper The Hasse-Brauer-Noether theorem in historical perspective is directly relevant to our subject matter.
Milne's notes on class field theory.
Gauss first proved the quadratic reciprocity law, leading to further developments of reciprocity laws by Eisenstein and Kummer. Hilbert in his 1897 Zahlbericht gave a profound analysis of Kummer's work leading to the notion of a class field. The main theorems of Class Field Theory were proved before 1920 by Takagi. Artin clarified the foundations by elucidating the special role of the Frobenius map, inventing the Artin L-functions, and giving lucid expositions of the theory. Meanwhile in 1932 Hasse, Brauer and Noether determined the structure of division algebras over local and global fields, allowing new foundations for class field theory. Ideles and Adeles were introduced by Chevalley and Weil, allowing a more streamlined treatment of the global theory. Finally Weil introduced the Weil group in 1951, and the role of Galois cohomology was clarified by Hochschild, Artin and Tate.

Last modified: Mon Jan 3 17:13:14 2005