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Let $q$ be a prime power and $n, r$ integers such that $r\mid q^n-1$. Anelement of $\mathbb{F}_{q^n}$ of multiplicative order $(q^n-1)/r$ is called\emph{$r$-primitive}. For any odd prime power $q$, we show that there exists a$2$-primitive element of $\mathbb{F}_{q^n}$ with arbitrarily prescribed$\mathbb{F}_q$ trace when $n\geq 3$. Also we explicitly describe the valuesthat the trace of such elements may have when $n=2$. A feature of this amendedversion is the reduction of the discussion to extensions of prime degree $n$.

The trace of 2-primitive elements of finite fields (amended version)