# Math 776

(under construction-subject to revision- 1 jan 2007).

Math 776, as the continuation of Math 676 , is a second-semester graduate course in algebraic number theory.
While Math 676 covered a variety of basic topics, Math 776 focuses on a single topic: Class Field Theory,
the study of abelian extensions of number fields (also, function fields over finite fields, and local fields).

## Course details

Time: MWF, 11AM-noon

Place: 3096 East Hall

First meeting: Friday, January 5, 2007

## About the instructor

Jeffrey Lagarias
Email: lagarias@umich.edu
Web: http://www.math.lsa.umich.edu/~lagarias/m776-0.html
Office: 3086 East Hall
Office Phone: 763-1186
Office Hours: TBA

## Course description

Class field theory, the study of abelian extensions of number fields, was a crowning achievement of number theory in the first half of the 20th century. It brings together, in a unified fashion, the quadratic and higher reciprocity laws of Gauss, Legendre et al, and vastly generalizes them. Some of its consequences (e.g., the Chebotarev density theorem) apply even to nonabelian extensions.

Our approach in this course will be to begin with review and the Kronecker-Weber theorem. Then we cover the formulations of the statements of global class field theory, for number fields, and local class field theory, for p-adic fields. Then I will follow one of:

Plan A.
Classical treatment as in Janusz, Global class field theory, via analysis. Local class field theory later.

Plan B. Study the cohomology of groups, an important technical tool both for class field theory and for many other applications in number theory. From there, we set up a local form of class field theory, then proceed to the main results.

## Textbooks

We will not follow a single text consistently. However, a good source is J.S. Milne's course notes on class field theory, which can be downloaded here.

A classical treatment is given in:

• G.Janusz, Algebraic Number Fields, 2nd Edition (AMS 2005)

A very good modern source is:

• J. Neukirch, Algebraic Number Theory (Springer-Verlag 1999-expensive).

Students are encouraged to have at least one of these last two sources available; I will point out as we go along where the material we cover in class is covered in both.

Other good references:

• Cassels and Frohlich, Algebraic Number Theory (excellent, but out of print)
• Serre, Local Fields (for local class field theory)
• Washington, Introduction to Cyclotomic Fields (for Kronecker-Weber, and some applications)

## Prerequisites

Math 676 or equivalent. Graduate coursework in algebra is also recommended.
If you are unsure how your background matches these prerequisites, see me.

## Homework/Exams

There will be problem sets approximately biweekly for most of the semester.

There will be no final exam; instead, students will submit a final paper (approx. 10 pages),
on a topic not covered during the course.

Some possible topics are listed below (under "Additional Topics"); feel free to propose your own topic!

## Course Topics: Tentative Syllabus

• Abelian extensions of the rationals: the Kronecker-Weber theorem (Milne, I.4; Washington, Chapter 14)
• Statements of results: classical form (Milne, V.1-V.3)
• Some applications of class field theory
• Statements of results: modern (adelic) form
• Group cohomology (Milne, II)
• Local class field theory (Milne, I.1 and III; Neukirch, IV and V)
• Proofs of the results of class field theory (Milne, VII; Neukirch, IV and VI)
• Additional topics (see below) if time permits.