UGA MATH 8330, Spring 2015. An introduction to the theory of finite dimensional division algebras, the Brauer group, involutions and other fun things.

dkrashen@uga.edu

- Wedderburn-Artin Theory
- Double commutators, Noether-Skolem
- Central simple algebras and central division algebras, degree and index
- Definition of the Brauer group
- Maximal subfields, crossed products
- Period, Index and the period-index problem
- 2-cocyles, basic Galois Cohomology
- Algebras with involutions, quadratic, symplectic and Hermitian forms
- Construction of classical algebraic groups
- Ramification, a taste of Azumaya algebras
- Examples and structure results (the Brauer group of local fields, the Albert-Hasse-Brauer-Noether Theorem and the Auslander-Brummer-Fadeev Theorems
- Polynomial identities (The Amitsur-Levitski Theorem and the Artin-Procesi Theorem)
- Generic division algebras and noncrossed product algebras
- Indecomposability and geometric methods
- Geometric Brauer groups, gerbes and obstructions