Besides being of interest in their own right, Central Simple Algebras are a basic tool in algebraic number theory (particularly in their interpretations via Galois cohomology), they arise naturally in algebraic geometry (as obstructions to the existence of universal objects in moduli problems), and also come up in representation theory over non-algebraically closed fields (in the construction of linear algebraic groups of classical type).

This course will assume comfort in 8000 level algebra, but will not assume additional knowledge beyond this. In particular, we will not assume a familiarity with algebraic geometry. I will, however, try to point out some connections with algebraic geometry, particularly during roughly the latter third of the course. The rough syllabus is:

– basic algebraic theory of central simple algebras

– construction of classical groups

– the Brauer group

– particular fields: local fields, number fields, transcendental extensions

– more modern methods, some connections with algebraic geometry