Existence results for primitive elements in cubic and quartic extensions of a finite field

Abstract

With $\Fq$ the finite field of $q$ elements, we investigate the following question. If ${\hat{I}}^3$ generates $\Fqn$ over $\Fq$ and ${\hat{I}}^2$ is a non-zero element of $\Fqn$, is there always an $a \in \Fq$ such that ${\hat{I}}^2({\hat{I}}^3+a)$ is a primitive element? We resolve this case when $n=3$, thereby proving a conjecture by Cohen. We also improve substantially on what is known when $n=4$.

To appear in Math. Comp