Pair of primitive elements in quadratic form with prescribed trace over a finite field

TL;DR

This paper provides a sufficient condition for the existence of primitive elements in finite fields with a prescribed trace, focusing on quadratic forms and identifying a unique exceptional case.

Contribution

It establishes a new criterion for primitive elements with prescribed trace in finite fields and characterizes the only exceptional case for certain parameters.

Findings

For all m ≥ 5, only one exceptional pair (q,m) = (2,6) exists.

Provides a sufficient condition for primitive elements with prescribed trace.

Analyzes quadratic forms with non-zero discriminant in finite fields.

Abstract

In this article, we establish a sufficient condition for the existence of primitive element $\alpha\in \Fm$ is such that $f(\alpha)$ is also primitive element of $\Fm$ and $Tr_{\Fm/\F}(\alpha)=\beta$, for any prescribed $\beta\in\F$, where $f(x)= ax^2 + bx + c\in \Fm(x)$ such that $b^2-4ac\neq 0$. We conclude that, for $m\geq 5$ there is only one exceptional pair $(q,m)$ which is $(2,6)$.