Taught Course Centre

Class Field Theory

January - March 2012.  Dr P-J White.

Description

Class field theory is an integral tool in number theory.  It provides a description of the abelian extensions of global fields (e.g. the rational

numbers) and local fields (e.g. the p-adic numbers).  For students interested in research, it also provides an entry point to the very active research area of the Langlands program.

 
In this course, we shall develop local class field theory via the cohomological approach.  If time permits, we shall briefly discuss global class field theory, and perhaps even the local Langlands program for GL(n) which generalises local class field theory.

Texts

There are many good textbooks on class field theory which treat the theory in different ways.  This course shall be based on the following two texts.

- Local Fields by Serre (or the French original "Corps Locaux")

- Algebraic Number Theory by Cassels and Frohlich (see the relevant chapters within).

Prerequisites

- A first course in commutative algebra.  That is knowledge to the level of "Introduction to Commutative Algebra" by Atiyah and Macdonald.

 The following are probably not strict prerequisites but are definitely good to know.

- Basic Galois theory.

- A first course in algebraic number theory (ideally some acquaintance with local fields).

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