1. Background

This is the second semester of algebraic number theory (ANT). The aim is to provide important tools and background knowledge in ANT. We will first continue the Fall Semester to complete Lang’s exposition of Global Class Field Theory (Artin reciprocity).

We then develop in details cohomology of groups and Galois cohomology, including Tate’s duality theorem. We also plan to cover the local approach of class field theory as well as the Lubin-Tate formal multiplication following Serre’s exposition. If time permits, we shall discuss Hilbert symbols, Chebotarev’s density theorem and Herbrand function.

2. Outline 

The course will consist of the following topics:

1.Recall the Artin reciprocity, Kummer theory, the existence theorem of class field theory, Hilbert class field and ray class field, Chebotarev’s density theorem

2. Cohomology of groups. Definition, Construction, functorial properties, Inflation-Restriction maps, Corestrion maps. Shapiro’s lemma, Tate’s theorem.

3. Galois cohomology, Central simple algebras, Brauer groups, and cohomological interpretation.

4. Local Class Field Theory, Lubin-Tate formal multiplication, the existence theorem.

5. Hilbert symbols, Herbrand’s function.

**Course extensions:

6/21 15:00-17:30
6/28 15:00-17:30

3. Credit: 3