Polynomials that represent quadratic residues at primitive roots

Daniel Joseph Madden and William Yslas Vélez

Vol. 98 (1982), No. 1, 123–137

DOI: 10.2140/pjm.1982.98.123

Abstract

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

Milestones

Received: 10 May 1980

Published: 1 January 1982