Not offered in 2017-18.

Nigel Boston (Spring 2017):

Class field theory is one of the great accomplishments of 20th century number theory. It describes all abelian extensions of every number field K in terms of the arithmetic of K, where "describes" means that you can also say how primes split/ramify. I shall begin by emphasizing explicit calculations and applications of this description. Then, with this as motivation, I shall prove the claimed description. My preference is to follow the proofs of Serre (local CFT) and Tate (global CFT) in the Cassels-Frohlich Brighton conference volume, but more detailed explanations can be found in Milne's online notes. I shall also discuss Galois cohomology, L-functions, and complex multiplication. Finally, I shall describe what we know and conjecture as regards nonabelian CFT.