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arXiv · 2202.00829

Primitive element pairs with a prescribed trace in the cubic extension of a finite field

Andrew R. Booker·Stephen D. Cohen·Nicol Leong·Tim Trudgian
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Abstract

We prove that for any prime power $q\notin\{3,4,5\}$, the cubic extension $\mathbb{F}_{q^3}$ of the finite field $\mathbb{F}_q$ contains a primitive element $\xi$ such that $\xi+\xi^{-1}$ is also primitive, and $\textrm{Tr}_{\mathbb{F}_{q^3}/\mathbb{F}_q}(\xi)=a$ for any prescribed $a\in\mathbb{F}_q$. This completes the proof of a conjecture of Gupta, Sharma, and Cohen concerning the analogous problem over an extension of arbitrary degree $n\ge3$.

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Andrew R. Booker, Stephen D. Cohen, Nicol Leong, Tim Trudgian. 2022-02-02. Primitive element pairs with a prescribed trace in the cubic extension of a finite field. https://arxiv.org/abs/2202.00829

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