On the existence of pairs of primitive and normal elements over finite fields

TL;DR

This paper establishes conditions under which finite field elements can be simultaneously primitive and normal, and also satisfy certain rational function properties, advancing understanding of element existence in finite fields.

Contribution

It provides a new sufficient condition for the existence of primitive, normal elements in finite fields that meet specific rational function criteria, extending previous results.

Findings

Conditions for the simultaneous primitiveness and normality of elements.

Existence of elements satisfying rational function constraints.

Theoretical framework for element existence in 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 $\textrm{deg}(f_i(x)) \leq m_i$, 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.