On the Existence of Pairs of Primitive and Normal Elements Over Finite Fields

Abstract

Let $$\mathbb {F}_{q^n}$$ be a finite field with $$q^n$$ elements, and let $$m_1$$ and $$m_2$$ be positive integers. Given polynomials $$f_1(x), f_2(x) \in \mathbb {F}_{q^n}[x]$$ with $$\deg (f_i(x)) \le m_i$$ , for $$i = 1, 2$$ , and such that the rational function $$f_1(x)/f_2(x)$$ satisfies certain conditions which we define, we present a sufficient condition for the existence of a primitive element $$\alpha \in \mathbb {F}_{q^n}$$ , normal over $$\mathbb {F}_q$$ , such that $$f_1(\alpha )/f_2(\alpha )$$ is also primitive.