Pair of primitive elements in quadratic form with prescribed trace over a finite field

In this article, we establish a sufficient condition for the existence of primitive element $α\in \Fm$ is such that $f(α)$ is also primitive element of $\Fm$ and $Tr_{\Fm/\F}(α)=β$, for any prescribed $β\in\F$, where $f(x)= ax^2 + bx + c\in \Fm(x)$ such that $b^2-4ac\neq 0$. We conclude that, for $m\geq 5$ there is only one exceptional pair $(q,m)$ which is $(2,6)$.

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