Cyclic algebras, symbol algebras and gradings on matrices

Abstract

We consider cyclic algebras, Milnor's symbol algebras, and certain graded algebra structures on them. We classify these gradings with respect to both isomorphism and equivalence relations. Some of them induce gradings on matrix algebras, which we also classify. As an application, we obtain the classification of all group gradings on the algebra Mp(F), where p is a prime number and F is an arbitrary field.