Matrix powers over finite fields

Abstract

Let GF(q) denote the finite field of order q=pe with p odd. Let M denote the ring of 2×2 matrices with entries in GF(q). Let n denote a divisor of q−1 and assume 2≤n and 4 does not divide n. In this paper, we consider the problem of determining the number of n-th roots in M of a matrix B∈M. Also, as a related problem, we consider the problem of lifting the solutions of X2=B over Galois rings.