Ma 160b:  Number Theory (Winter 2016-17)

## ANNOUNCEMENTS

Registration for Winter term opens Thursday, November 17, 2016.

## COURSE DESCRIPTION

The focus of this course is local class field theory, to be followed in the spring term by its companion course Math 160c covering global class field theory. We will study local fields, their ramification and Galois groups, Galois cohomology, Brauer groups, local duality, the invariant map, the Euler characteristic formula and possibly explicit class field theory in Lubin-Tate towers and local Galois representations. The course assumes a working knowledge of abstract algebra, Galois theory and basic algebraic number theory.

## PREREQUISITES

Math 160a and Math 120b are necessary.

## SCHEDULE

MWF 14:00 - 14:55, 257 Sloan.

## INSTRUCTORS

Zavosh Amir Khosravi
Olga Taussky and John Todd Instructor in Mathematics
Sloan 258

## OFFICE HOURS

Wednesdays 1-2pm, Sloan 258

## POLICIES

Each student in the class is asked to give a final presentation on a topic/result/computation related to the material covered by the class.

Here is a list of sample topics:

Proof of the Hasse-Arf Theorem;

The local Kronecker-Weber theorem by elementary methods;

Hilbert symbols and a proof of LCFT existence theorem;

Artin representations (an example);

The Brauer group of local field (classical definition and results via Central simple algebras);

Global class field theorem, an outline of the proof GCFT starting from LCFT via Tate cohomology;

The Grunwald-Wang theorem (and or other cases of local-global principles);

The ramifications subgroups for the LT tower.

The cup product on group cohomology.

## TOPICS COVERED

(Complete) Discrete Valuation Fields

Ramification Groups

Galois Cohomology

Brauer Groups of Local Fields

Lubin-Tate Theory

Local Class Field Theory -- Existence Theorem

## TEXTBOOKS

The main textbook for the course is:

Local Fields by Jean-Pierre Serre, GTM Springer

Milne's class field theory notes.

Keith Conrad's history of class field theory.

The original Lubin-Tate paper.

Date Description
Jan 13 Notes
Jan 18 Notes
Jan 20 Notes
Jan 23 Notes
Jan 25 Notes

## HOMEWORK

Due Date Homework Solutions
Jan. 20 HW1
Feb. 3 HW2
Mar. 3 HW3