is a purely transcendental extension of k. Much has been written on this topic for certain groups G, particularly abelian groups or linear groups over fields of characteristic zero (see, for example, [4, 5, 6]) but, on the whole, very little is known when G is non-abelian especially over fields of positive characteristic. In this paper we answer the question affirmatively for a case of the orthogonal group over a finite field. Let Fq be the finite field of odd prime power order q. Given a non-degenerate quadratic form Qn(xv...,xn) on V = F%, the orthogonal group O(n, Qn) is the set of linear transformations a on V such that Qn{av) = Qn(v) for all v in V. While, in principle, Qn is arbitrary, in practice, as is well known, Qn can be taken to be a diagonal form and indeed can be specified uniquely when n is odd and as one of two alternative forms when n is even (see [3, §§6.3, 6.10] and below). The group O(«, Qn), a subgroup of the general linear group GLn(Fq), acts as a group of /^-automorphisms on the polynomial ring Rn = Fq[xv...,xn] and its associated quotient field Kn = FQ(x19...,xn) in a natural way. The invariant subfield A^» will be denoted by Kn and the corresponding invariant subring of Rn by /?+. Evidently, R^ = Rnft K+ and Rn is the integral closure of R+ in Kn [1, p. 68, Exercise 12]. In this situation, Huah Chu [2] has conjectured that K^ is purely transcendental in precise terms, for whose statement we introduce some notation. For a non-negative integer i and a diagonal form Qn put