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).

Place: 3096 East Hall

First meeting: Friday, January 5, 2007

Email:

Web:

Office: 3086 East Hall

Office Phone: 763-1186

Office Hours: TBA

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.

A classical treatment is given in:

A very good modern source is:

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)

If you are unsure how your background matches these prerequisites, see me.

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!

- 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.

- The Lubin-Tate approach to local class field theory (Milne, I.2)
- Brauer groups (Milne, IV)
- Quadratic forms over number fields (Milne, IV)
- The Carlitz module and class field theory for function fields (David Hayes, A brief introduction to Drinfeld modules, in
*The Arithmetic of Function Fields*) - Serre's approach to class field theory for function fields (Serre,
*Algebraic Groups and Class Fields*) - Class field towers: the Golod-Shafarevich inequality (Cassels-Frohlich)
- Complex multiplication: explicit class field theory for imaginary quadratic fields (Neukirch, VI.6)
- Zeta functions and number fields (Neukirch, VII)
- Computations of Hilbert class fields

Jeff Lagarias (lagarias(at)umich.edu)