# Primitive Element Pairs with a Prescribed Trace in the Quartic Extension of a Finite Field

## TL;DR

This paper proves that in the quartic extension of a finite field, there exists a primitive element with a prescribed trace such that its sum with its inverse is also primitive, extending known results for higher degrees.

## Contribution

It provides a self-contained proof for the existence of primitive elements with a prescribed trace and specific properties in quartic finite field extensions.

## Key findings

- Existence of primitive elements with prescribed trace in $_{q^4}$
- Primitive element $eta$ with $eta+eta^{-1}$ also primitive
- Extension of known results to degree 4 case

## Abstract

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.