  In this paper we generalize the results of Sharma, Awasthi and Gupta (see
\cite{SAG}). We work over a field of any characteristic with $q = p^k$ elements
and we give a sufficient condition for the existence of a primitive element
$\alpha \in \mathbb{F}_{p^k}$ such that $f(\alpha)$ is also primitive in
$\mathbb{F}_{p^k}$, where $f(x) \in \mathbb{F}_{p^k}(x)$ is a quotient of
polynomials with some restrictions. We explicitly determine the values of $k$
for which such a pair exists for $p=2,3,5$ and $7$.
