  In this article, we give a largely self-contained proof that the quartic
extension $\mathbb{F}_{q^4}$ of the finite field $\mathbb{F}_q$ contains a
primitive element $\alpha $ such that the element $\alpha+\alpha^{-1}$ is also
a primitive element of ${\mathbb{F}_{q^4}},$ and
$Tr_{\mathbb{F}_{q^4}|\mathbb{F}_{q}}(\alpha)=a$ for any prescribed $a \in
\mathbb{F}_q$.
  The corresponding result for finite field extensions of degrees exceeding 4
has already been established by Gupta, Sharma and Cohen.
