Math 254B Syllabus
Last revised: 17 Nov 2001.
Math 254B, as the continuation of Math 254A, is a second-semester
graduate course in algebraic number theory. While Math 254A covered a variety
of basic topics, Math 254B will focus on a single topic: class field theory,
the study of abelian extensions of number fields.
Time: Mon, Wed, Fri 2-3 PM
Place: 4 Evans
First meeting: Wednesday, January 23
About the instructor
Kiran S. Kedlaya
Office: 757 Evans
Office Phone: 642-6923
Office Hours: Wed/Thurs 11 AM-noon, or by appointment, or just drop by!
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 the formulations of
the statements of class field theory, omitting the proofs (except
for the Kronecker-Weber theorem, which we prove first). We then proceed
to 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.
We will not follow a single text consistently. However, Neukirch's
Algebraic Number Theory, the text for Math 254A, is a wonderful
and thorough resource; another good source is J.S. Milne's course notes
on class field theory, which can be downloaded
Students are strongly encouraged to have at least
one of these 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)
Math 254A or equivalent. Graduate coursework in algebra
(Math 250 or equivalent) is also recommended. If you are unsure how your
background matches these prerequisites, see me.
There will be problem sets approximately weekly for most of the semester.
There will be no final exam; instead, students will submit a final paper
on a topic not covered during the course. For more details, see
the guidelines. Some suggested topics are listed
below (under "Additional Topics"); feel free to come up with others!
For a more detailed breakdown, see the
course notes page.
Abelian extensions of the rationals: the Kronecker-Weber theorem
(Milne, I.4; Washington, Chapter 14)
Statements of results: classical form
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.
These are topics that I do not plan to cover in the course, but should be
easily accessible once we have completed the planned syllabus. In particular,
these topics would be suitable for a final paper. This list is subject to
change as the semester progresses; a definitive version will be distributed
at about the middle of the semester.
- 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 functions 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)
Kiran S. Kedlaya