Invariants of 2×2-Matrices over Finite Fields

Let q be the finite field with q elements, q=pν, p a prime, and Mat2.2( q) the vector space of 2×2-matrices over . The group GL(2, ) acts on Mat2,2( q) by conjugation. In this note, we determine the invariants of this action. In contrast to the case of an infinite field, where the trace and determinant generate the ring of invariants, several new invariants appear in the case of finite fields.