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 in algebraic number theory,

Math 776 traditionally focuses on a single topic: Class Field Theory,
the study of abelian extensions of number fields.

Place: 1866 East Hall

First meeting: Thursday, January 5, 2017

Email:

Web:

Office: 3086 East Hall

Office Phone: 763-1186

Office Hours: TBD or by appointment.

relevant to understanding parts of S. Mochizuki's IUT theory. A user-friendly approach to that theory would use the text

This book is different from Neukirch's other books on
algebraic number theory and class field theory. It give an abstract version of class field theory,

encoding it
as purely group-theoretic constructions, and treats infinite extensions of algebraic number fields, starting at page 1.

It gets to the main results quickly, but is
opaque if you never saw class field theory before,
and even if you have.

I am considering to try it.

* Serge Lang,Algebraic Number Theory, 2nd Edition * Springer-Verlag 1994.

This text seems a bit dry and unmotivated, but it covers the
right things on the analytic side.

A second very

good source is J.S. Milne's course notes
on class field theory, which can be downloaded
here.

A classical source is

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

A very good modern source is:

J. Neukirch, *Algebraic Number Theory [ANT] * (Springer-Verlag 1999).

Students are encouraged to have at least one of these last two sources available.

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.

This approach in this course will be to begin with review and then to prove the Kronecker-Weber theorem (class field theory over the rational field). Then we cover Kummer theory, the formulations of the statements of global class field theory, for number fields, and local class field theory, for p-adic fields. Then it does other stuff.

- 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 [ANT], IV and V)
- Proofs of the results of class field theory (Milne, VII; Neukirch [ANT], IV and VI)
- Additional topics as time permits.

Jeff Lagarias (lagarias(at)umich.edu)