Some Four-Dimensional Orthogonal Invariants

Let [Formula: see text] be an odd prime and [Formula: see text] be the prime field of order [Formula: see text]. Consider a [Formula: see text]-dimensional orthogonal group [Formula: see text] over [Formula: see text] acting on the standard representation [Formula: see text] and the dual space [Formula: see text]. We compute the invariant ring [Formula: see text] via explicitly exhibiting a minimal generating set. Our method provides an application of the [Formula: see text]-invariants appearing in the covariant theory of finite groups.