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By definition primitive and $2$-primitive elements of a finite fieldextension $\mathbb{F}_{q^n}$ have order $q^n-1$ and $(q^n-1)/2$, respectively.We have already shown that, with minor reservations, there exists a primitiveelement and a $2$-primitive element $\xi \in \mathbb{F}_{q^n}$ with prescribedtrace in the ground field $\mathbb{F}_q$. Here we amend our previous proofs ofthese results, firstly, by a reduction of these problems to extensions of primedegree $n$ and, secondly, by deriving an exact expression for the number ofsquares 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.

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