On r-primitive k-normal elements with prescribed norm and trace over finite fields

Abstract

Let q=pm, where p is a prime and m∈N. Let Fqn be the field extension of degree n over Fq. Further, let r≥1 and k≥0 be integers such that r|qn−1 and there exists a k degree polynomial g(x)|xn−1. Let ξ∈Fq⁎ be an ω-primitive element i.e. the multiplicative order of ξ is (qn−1)/ω, where ω=gcd(r,q−1), and ζ∈Fq be any arbitrary element.In this article, we establish a sufficient condition for the existence of an r-primitive k-normal element α∈Fqn such that NFqn/Fq(α)=ξ and TrFqn/Fq(α)=ζ. Using this sufficient condition, we show that, for fixed integers r and k there always exists an r-primitive k-normal element with prescribed norm ξ and trace ζ in all but finitely many finite fields Fqn, for n≥4k+5.