Polynomials that represent quadratic residues at primitive roots
Daniel J. Madden, William Vélez · Pacific Journal of Mathematics · 1982
In this paper the following result is obtained.THEOREM.Let r be any positive integer; in all but finitely many finite fields k, of odd characteristic, for every polynomial f(x) e k[x] of degree r that is not of the form a(Q(x)) 2 or ax(9(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 shall see that for every finite field k of characteristic Φ 2 or 3, there exists a primitive root aek such that -(a 2 + a + 1) = β 2 for some βek; also every linear 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(S), GF(S) or GF(Ί).