  We describe the ring of invariants for the finite orthogonal group of plus
type in odd characteristic acting on its defining representation. We also
describe the invariants of its Sylow subgroup in the defining characteristic.
In both cases we construct minimal algebra generating sets and describe the
relations among the generators. Both rings of invariants are shown to be
Complete Intersections and thus are Cohen-Macaulay. We expect the techniques we
use will generalise to give a systematic computation for rings of invariants
for all of the finite classical groups.
