Vol. 98, No. 1, 1982

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Polynomials that represent quadratic residues at primitive roots

Daniel Joseph Madden and William Yslas Vélez

Vol. 98 (1982), No. 1, 123–137

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) k[x] of degree r that is not of the form α(g(x))2 or αx(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 shall see that for every finite field k of characteristic 2 or 3, there exists a primitive root α k such that (α2 + α + 1) = β2 for some β k; 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(3), GF(5) or GF(7).

Mathematical Subject Classification
Primary: 12C05, 12C05
Received: 10 May 1980
Published: 1 January 1982
Daniel Joseph Madden
William Yslas Vélez