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arXiv · 2007.09787

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

Cícero Carvalho·João Paulo Guardieiro·Victor G. L. Neumann·Guilherme Tizziotti
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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.

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Cícero Carvalho, João Paulo Guardieiro, Victor G. L. Neumann, Guilherme Tizziotti. 2020-07-19. On the existence of pairs of primitive and normal elements over finite fields. https://arxiv.org/abs/2007.09787

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