Primitive Element Pairs with One Prescribed Trace over a Finite Field

TL;DR

This paper proves the existence of primitive elements in finite fields with specific properties, including prescribed trace and the element's inverse sum also being primitive, for most field sizes and extensions.

Contribution

It establishes a sufficient condition for such primitive elements to exist in finite fields, extending previous results and eliminating exceptions for large extension degrees.

Findings

Almost all finite fields of degree n≥5 contain such primitive elements.

No exceptional pairs (q,n) found for n≥5 through computation.

Provides a criterion for the existence of primitive elements with prescribed trace and inverse sum properties.

Abstract

In this article, we establish a sufficient condition for the existence of a primitive element $\alpha \in {\mathbb{F}_{q^n}}$ such that the element $\alpha+\alpha^{-1}$ is also a primitive element of ${\mathbb{F}_{q^n}},$ and $Tr_{\mathbb{F}_{q^n}|\mathbb{F}_{q}}(\alpha)=a$ for any prescribed $a \in \mathbb{F}_q$, where $q=p^k$ for some prime $p$ and positive integer $k$. We prove that every finite field $\mathbb{F}_{q^n}~ (n \geq5),$ contains such primitive elements except for finitely many values of $q$ and $n$. Indeed, by computation, we conclude that there are no actual exceptional pairs $(q,n)$ for $n\geq5.$