Franz Lemmermeyer

Class Field Theory

Class field theory is a branch of algebraic number theory that studies abelian extensions of number fields. Over the rationals, for example, the theorem of Kronecker-Weber states that every abelian extension is contained in some cyclotomic field (a number field generated by a suitable root of unity). Since class field theory describes these abelian extensions in terms of generalized class groups, it is the principal tool for proving theorems about ideal class groups. My approach will be classical (this means: (almost) no Galois cohomology, no local class field theory, no ideles; I will, however, occasionally point out connections to these objects). Prerequisites are a good command of algebraic number theory (unique factorization into prime ideals, ideal class groups, units) and Galois theory.
It is almost impossible to cover class field theory in one semester (except by using all the fancy tools mentioned above), so occasionally I might have to sketch a proof or two, in particular since I also would like to briefly sketch a few important topics, such as I intend to vaguely follow lecture notes by Hasse and Iyanaga, with bits and pieces from the book "Algebraic Number Theory" by Janusz. Other books on class field theory are by Artin-Tate, Cassels-Frölich, and Neukirch, but they all use local class field theory, as well as cohomology (at least to a certain extent). Here is the first part of Hasse's Lectures on Class Field Theory. His introduction to Galois theory is hopelessly outdated, but the last part on Frobenius and Artin symbols is quite readable.


You're allowed (and even encouraged) to work in groups and discuss the problems; the final draft, however, should be done independently by each student.


ok - here's the most recent version of the notes (updated April 30; apparently that the gaps are growing instead of doing me the favor of disappearing . . . )
Noah Snyder's historical account of Artin's L-functions can be found here.
Anatoly Preygel has written a paper on L-functions and density of primes.
Here are a few expositions of parts of Janusz's book by Daileda and Goins.
Here's a nice article by Lenstra and Stevenhagen on Chebotarev and his density theorem.

What I did in class