Polynomials of degree 4 over finite fields representing quadratic residues

  • Shaofei Du Capital Normal University, China
  • Klavdija Kutnar University of Primorska, Slovenia https://orcid.org/0000-0002-9836-6398
  • Dragan Marušič University of Primorska, Slovenia, and Institute of Mathematics, Physics and Mechanics, Slovenia
Keywords: Finite field, polynomial, quadratic residues


It is proved that in a finite field F of prime order p, where p is not one of finitely many exceptions, for every polynomial f(x) ∈ F[x] of degree 4 that has a nonzero constant term and is not of the form αg(x)2 there exists a primitive root β ∈ F such that f(β) is a quadratic residue in F. This refines a result of Madden and Vélez from 1982 about polynomials that represent quadratic residues at primitive roots.