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arXiv · 2108.08066

The trace of primitive and $2$-primitive elements in finite fields, revisited

Stephen D. Cohen·Giorgos Kapetanakis
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Abstract

By definition primitive and $2$-primitive elements of a finite field extension $\mathbb{F}_{q^n}$ have order $q^n-1$ and $(q^n-1)/2$, respectively. We have already shown that, with minor reservations, there exists a primitive element and a $2$-primitive element $\xi \in \mathbb{F}_{q^n}$ with prescribed trace in the ground field $\mathbb{F}_q$. Here we amend our previous proofs of these results, firstly, by a reduction of these problems to extensions of prime degree $n$ and, secondly, by deriving an exact expression for the number of squares in $\mathbb{F}_{q^n}$ whose trace has prescribed value in $\mathbb{F}_q$. The latter corrects an error in the proof in the case of $2$-primitive elements. We also streamline the necessary computations.

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Stephen D. Cohen, Giorgos Kapetanakis. 2021-08-18. The trace of primitive and $2$-primitive elements in finite fields, revisited. https://arxiv.org/abs/2108.08066

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