Polynomials that represent quadratic residues at primitive roots

In this paper the following result is obtained. THEOREM. Let r be any positiveinteger; in all but finitely many finite fields k, of odd characteristic, for every polynomial f(x) ∈ k[x] of degree r that is not of the form ∝(g(x))² or ∝x(g(x))², there exists a primitive root β ∈ k such that f(β) is a square in k. As a result of this and some computation we shallsee that for every finite field k of characteristic ≠ 2 or 3, there exists a primitive root β ∈ k such that — (∝² + ∝ + 1) = β² for some ek; also everylinear polynomial with nonzero constant term in the finite field k of odd characteristic represents both nonzero squares and nonsquares at primitive roots of k unless k = GF(3), GF(5) or GF(7).