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

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.

arXiv admin note: substantial text overlap with arXiv:1903.03160