Matrices over maximal orders in cyclic division algebras over Q as sums of squares and cubes

Abstract

It is known that every matrix of order n over the maximal order in an algebraic number eld is a sum of k-th powers in various cases if a discriminant condition is satis ed. It has been proved by Wadikar and Katre that for every matrix of size 2 over maximal orders in rational quaternion division algebras is a sum of squares and cubes. In this paper we consider cyclic division algebras over Q of odd prime degree and show that under some conditions every matrix of size greater equal 2 over these noncommutative rings is a sum of squares and a sum of cubes.