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This course will aim to give an introduction to the Brauer group of a field, an object ubiquitous in modern algebraic number theory. The first half of the course will introduce the Brauer group via the classical theory of central simple algebras (a certain class on noncommutative rings of which Hamilton's quaternions form a key example). Along the way we will develop various tools in noncommutative algebra. The second half of the course will develop the basic theory of group cohomology and use this to give an alternative description of the Brauer group.
Lectures 1-5: Definitions and examples of central simple algebras, Wedderburn's theorem, splitting fields for central simple algebras and introduction to the Brauer group of a field.
Lectures 6-8: Cohomology of finite and profinite groups.
Lectures 9-10: Galois descent and cohomological description of the Brauer group.
Possible additional topics (time permitting) include rational representations and Schur indices, computation of the Brauer group of the p-adic numbers, and description of the Brauer group of a number field.
References: The course will be accompanied with a full set of typed notes, which will be updated regularly as the course progresses. The main reference for the first half of the course is the book of the same name: `Central Simple Algebras and Galois Cohomology' by Philippe Gille and Tamas Szamuely. This also contains much of the material of the second half of the course, but for the lectures on group cohomology we will follow much more closely chapter IV (`Cohomology of Groups', M.F. Atiyah and C.T.C. Wall) of the book `Algebraic Number Theory' by Cassels and Frohlich. This course should dovetail well with the core module `Algebras and Representation Theory' which is running concurrently this semester. The first halves of each course in particular will have several notions in common.
Prerequisites: Familiarity with commutative algebra (groups, rings, modules, tensor products) and some previous exposure to Galois theory will be very helpful. Basics of representation theory and homological algebra will be useful but not essential.