Math 845 - Spring 2017

Nigel Boston

Contact Information

303 Van Vleck Hall.

Telephone: 263-4753.



Office Hours: In 3619 EH, 1:30-3 Mon; in 303 VV, 1:30-2:30 Th and 3-3:30 F; or by appointment.


There is no perfect text for class field theory. The best resources for teaching it are, I think, Milne's notes and Neukirch's book ``Algebraic Number Theory". I was planning on combining these, but then realized that Kedlaya's notes do this already. I will therefore be mostly following his notes - check them out! I learned Class Field Theory from the masterful articles of Serre and Tate in the Brighton volume. If you want to get to the heart of matters fast, it's hard to beat these. See the scan below.

Course Overview

Class field theory is the description of extensions of a number field (or local field) K in terms of the arithmetic of K. For extensions with abelian Galois group, the theory was the focal point of algebraic number theory from about 1850 to 1930. The nonabelian case has many conjectures and increasingly many proofs (in fact the last decade or so has seen exciting advances). In this course abelian class field theory will be completely covered, requiring the introduction of many of the tools in the armory of the modern number theorist, such as Galois cohomology, L-series, etc. Applications of historical and modern importance will be presented en route together with several concrete examples. Check out the course notes to learn more. There is a lovely article by Barry Mazur on constructing abelian extensions of number fields that continues from around about where this course will end.


Homework Assignments

Some Lists of Unsolved Problems

General Information (e.g. Conferences)

Online Notes

Books on Class Field Theory

History of Class Field Theory

Computational Resources