Primitive element pairs with a prescribed trace in the cubic extension of a finite field

TL;DR

This paper proves that for most finite fields, there exists a primitive element in the cubic extension with a prescribed trace and whose sum with its inverse is also primitive, confirming a conjecture for degree three extensions.

Contribution

The paper establishes the existence of primitive elements with prescribed trace and inverse properties in cubic extensions, completing a conjecture for all degree three cases.

Findings

Existence of primitive elements with prescribed trace in cubic extensions.

Primitive element whose inverse sum is also primitive exists in most finite fields.

Confirms a conjecture for degree three extensions of finite fields.

Abstract

We prove that for any prime power $q\notin\{3,4,5\}$, the cubic extension $\mathbb{F}_{q^3}$ of the finite field $\mathbb{F}_q$ contains a primitive element $\xi$ such that $\xi+\xi^{-1}$ is also primitive, and $\textrm{Tr}_{\mathbb{F}_{q^3}/\mathbb{F}_q}(\xi)=a$ for any prescribed $a\in\mathbb{F}_q$. This completes the proof of a conjecture of Gupta, Sharma, and Cohen concerning the analogous problem over an extension of arbitrary degree $n\ge3$.