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**1 - 2**of**2**### WARING’S PROBLEM FOR MATRICES OVER ORDERS IN ALGEBRAIC NUMBER FIELDS

, 2007

"... Abstract. In this paper we give necessary and sufficient trace conditions for an n×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 ≥ 2 are integers. We prove a discriminant criterion for every 2×2 matrix over an order R ..."

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Abstract. In this paper we give necessary and sufficient trace conditions for an n×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 ≥ 2 are integers. We prove a discriminant criterion for every 2×2 matrix over an order R in an algebraic number field to be a sum of cubes and fourth powers of matrices over R. We also show that if q is a prime and n ≥ 2, then every n × n matrix over the ring O of integers 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 field.