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.

Course details

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!

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 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 here. 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:


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.

Additional topics

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.
Kiran S. Kedlaya (kedlaya(at)