BRAUER ALGEBRAS AND THE BRAUER GROUP

Abstract. An algebra is a vector space V over a field k together with a k-bilinear product of vectors under which V is a ring. A certain class of algebras, called Brauer algebras- algebras which split over a finite Galois extension-appear in many subfields of abstract algebra, including K-theory and class field theory. Beginning with a definition of the the tensor product, we define and study Brauer algebras, and in particular their connections with Galois cohomology and algebraic geometry. Key results include the fact that Brauer algebras are precisely the finite dimensional central simple algebras; that the Brauer algebras under the right equivalence relation form an abelian group; that this group can be identified with a first (and second) cohomology set; an