Polynomial invariants of classical subgroups of GL2: Conjugation over finite fields

Consider the conjugation action of the general linear group GL₂(K) on the polynomial ring K[X2 × 2]. When K is an infinite field, the ring of invariants is a polynomial ring generated by the trace and the determinant. We describe the ring of invariants when K is a finite field, and show that it is a hypersurface. We also consider the other classical subgroups, and the polynomial rings corresponding to other subspaces of matrices such as the traceless and symmetric matrices. In each case, we show that the invariant ring is either a polynomial ring or a hypersurface.