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 $f_1(x),f_2(x)\in {\mathbb{F}}_q[x]$ with $\text{deg}(f_i(x))\le m_i$, for $i=1,2$, and such that the rational function $f_1(x)/f_2(x)$ belongs to a certain set 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.