Title: 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 $\gamma$ generates $\Fqn$ over $\Fq$ and $\beta$ is a non-zero
element of $\Fqn$, is there always an $a \in \Fq$ such that $\beta(\gamma + a)$
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 $n=4$.