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

TL;DR

This paper investigates the existence of primitive elements in finite field extensions, proving a conjecture for cubic extensions and improving results for quartic extensions.

Contribution

It proves Cohen's conjecture for cubic extensions and advances the understanding of primitive elements in quartic extensions of finite fields.

Findings

Confirmed existence of primitive elements in cubic extensions as conjectured by Cohen.

Significantly improved results on primitive elements in quartic extensions.

Provided new methods for analyzing primitive elements in finite field extensions.

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 $n=4$.