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MA6M3 Local Fields

Lecturer: Christopher Williams

Term(s): Term 2

Status for Mathematics students: 

Commitment: 30 lectures

Assessment: 85% by 3-hour examination and 15% coursework

Formal registration prerequisites: None

Assumed knowledge:

  • MA260 Norms, Metrics and Topologies (or MA222 Metric Spaces): I will freely use material from this entire course, with a vital role played by norms, their associated metrics, and topological spaces, including notions such as compactness and completeness.
  • MA3A6 Algebraic Number Theory: material from the first half of the course will be essential, particularly including: algebraic number fields, rings of algebraic integers, norms and traces, prime ideals and their factorisation in extensions of number fields.
  • MA3D5 Galois Theory: (finite) field extensions, Galois extensions and Galois groups. The theory of finite fields (and their Galois extensions) will be particularly crucial.

Useful background: MA3G6 Commutative Algebra: this course gives a thorough (and general) background to concepts such as valuations, local rings and localisation. These will all be defined/recalled during the Local Fields course, but some familiarity with these concepts will be useful.

Synergies: Local Fields is a topic in algebraic number theory. Whilst there will not be direct overlap in the course material, on a deeper level MA426 Elliptic Curves and MA4H9 Modular Forms are strongly related courses, and all three are likely to be useful for students looking to pursue future study (e.g. a PhD) in algebraic number theory (or, for example, an MA4K9 Research Project in this area). This module will also go well with MA4J8 Commutative Algebra II, which will treat topics useful for studying Local Fields in a general abstract context.

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.