(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).
Time: MWF, 11AM-noon
Place: 3096 East Hall
First meeting: Friday, January 5, 2007
About the instructor
Office: 3086 East Hall
Office Phone: 763-1186
Office Hours: TBA
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:
Classical treatment as in Janusz, Global class field theory,
via analysis. Local class field theory later.
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,
a good source is J.S. Milne's course notes
on class field theory, which can be downloaded
A classical treatment is given in:
Algebraic Number Fields, 2nd Edition (AMS 2005)
A very good modern source is:
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)
Math 676 or equivalent. Graduate coursework in algebra
is also recommended.
If you are unsure how your
background matches these prerequisites, see me.
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
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 which should be
accessible once we have completed the planned syllabus.
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
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 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