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).
• Hilbert's theory of ramification (Galois theory applied to the decomposition of prime ideals)
• The Frobenius automorphism and the Artin symbol
• Zeta functions and L-series
• Frobenius density theorem
• The first inequality
• Herbrand's Lemma and a little bit of cohomology
• The second inequality
• Artin reciprocity
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
• Artin L-functions and representation theory
• Galois representations
• class field towers
• principal ideal theorem
• reciprocity laws
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 His introduction to Galois theory is hopelessly outdated, but the last part on Frobenius and Artin symbols is quite readable.

### Homework

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.
• Th 01.02.: problems 1.3, 1.4, 1,6, 1.10, 1.11
• Tu 13.02.: problems 2.1, 2.4, 2.5, 2.6, 2.7
• Tu 06.03.: problems 1.9 - 1.13 from the new chapter 1.
• Tu 20.03.: problems 3.1 - 3.4
• We 04.04.: problems 6.2, 6.6, 6.7
• Tu 01.05.: problems 11.5 - 11.9

### Notes

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

• Tu 30.01. Review of Algebraic Number Theory
• Th 01.02. Review of Algebraic Number Theory
• Tu 06.02. Galois Extensions and Ramification Groups; Artin symbol
• We 07.02. 13:40 - 14:30 Quadratic Reciprocity; Hilbert Class Fields
• Tu 13.02. Hilbert class fields, Generalized class groups
• We 14.02. No class; moved to 28.02
• Tu 20.02. Generalized class groups; analytic methods
• We 21.02. Riemann zeta function; infinitude of primes
• Tu 27.02. Dirichlet's computation of L(1); Dirichlet density
• We 28.02. Quadratic class number formula I; homework
• Tu 06.03. Class number formula for quadratic fields II.
• We 07.03. Dirichlet's Theorem on Primes in Arithmetic Progression
• Tu 13.03. Nonvanishing of L-series at s=1
• We 14.03. Factorization of the zeta function of cyclotomic fields
• Tu 20.03. Minkowski bounds; Dirichlet's unit theorem
• We 21.03. Dedekind's class number formula
• Tu 27.03. Dedekind's class number formula
• We 28.03. Prime ideal decomposition in nonnormal extensions
• Tu 03.04. Frobenius density theorem
• We 04.04. First Inequality
• spring break
• Tu 17.04. Preparations for the second inequality
• Th 18.04. Index calculations
• Tu 24.04. Herbrand quotient of units
• Th 25.04. Herbrand's unit theorem
• Tu 01.05. Main Theorems of Class Field Theory
• Th 02.05. Examples of Hilbert Class Fields
• Tu 08.05. Artin reciprocity and the existence theorem (sketch)
• Th 09.05. Norm residues and Local class field theory; ideles (sketch)