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

TL;DR

This paper revisits and refines previous results on the existence of primitive and 2-primitive elements in finite fields with prescribed trace, providing exact counts and correcting earlier errors.

Contribution

It amends prior proofs by reducing problems to prime degree extensions and deriving exact formulas for squares with prescribed trace in finite fields.

Findings

Corrected proof for the existence of primitive and 2-primitive elements with prescribed trace

Derived an exact expression for counting squares with a given trace in finite fields

Streamlined computational methods for these problems

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.