On polynomials of small degree over finite fields representing quadratic residues

Abstract

Let F be a finite field of odd characteristic. It was proved by Madden and Vélez in [Pacific J. Math. 98(1982), 123–137] that except for finitely many exceptions of F, for every polynomial f(x) ∈ F[x] which is not of the form αg(x)2 or αxg(x)2, where α ∈ F, there exists a primitive root β ∈ F such that f(β) is a nonzero square in F. When this theoretical result is concretely used, it is necessary to determine such finitely many exceptions. In this paper, the possible exceptions are determined, provided the degree of f(x) is less than 9. Remark that our arguments do not hold when f(x) is of degree 9 and also the capacity at this stage of a computer is not enough to do it. Moreover, such kinds of results have been used to solve some combinatorial problems.