  In this paper we give necessary and sufficient trace conditions for an n by n
matrix over any commutative and associative ring with unity to be a sum of k-th
powers of matrices over that ring, where n,k are integers greater equal 2. We
prove a discriminant criterion for every 2 by 2 matrix over an order R to be
sums of cubes and fourth powers over R.
  We also show that if q is a prime and n greater equal 2, then every n by n
matrix over the ring of integers O, of a quadratic number field is a sum of
q-th powers (of matrices) over O if and only if q is coprime to the
discriminant of the quadratic number field.

Number Theory