Content: The real numbers R are defined as the completion of the rational numbers Q in the usual metric. However, this metric is not that well-suited to arithmetic study; for example, the integers are discrete in R.

In number theory, one is often more interested in p-adic numbers Qp, the completion of Q in the p-adic metric. In the p-adic metric, a number is very close to zero if it is highly divisible by a prime p (for example, whilst 1,000,000,000 is ‘large’ in the usual metric, it is highly divisible by 2 and 5, so it is very small in the 2-adic and 5-adic metrics). The integers are not discrete in the p-adic metric (as e.g. one can arbitrarily approximate 0 by integers p-adically), so p-adic numbers are much better suited to arithmetic, and have accordingly become fundamental in number theory and arithmetic geometry.

The real and p-adic numbers are examples of local fields. This module will give an introduction to local fields, with an emphasis on the p-adic numbers/non-archimedean local fields, and describe some of their beautiful properties, including: the classification of local fields, Hensel’s lemma and applications to solubility of polynomials, and extensions and Galois theory of local fields.

The course will also treat some notable applications in number theory and arithmetic geometry, in particular the Kronecker—Weber theorem on abelian extensions of Q and the Hasse—Minkowski theorem on solubility of quadratic forms.

Aims: To give students a grounding in the theory of local fields (e.g. the p-adic or real numbers) and their relationship with global fields (e.g. the rationals), and to gain insight into the use of local methods to solve global problems.

Objectives: By the end of the module, students should be able to:

• Explain the definition, basic properties and classification of valuations and local fields
• Understand inverse limits and the topology of the p-adic integers
• Use Hensel’s lemma to determine solubility of polynomial equations over local fields
• Use the Hasse—Minkowski theorem to determine solubility of rational quadratic forms
• Describe the Galois theory of local fields, including solubility of the Galois group and classification of abelian extensions

Books: Primary resources will include the books Local Fields by Serre and Local Fields by Cassels.