The translate and line properties for 2-primitive elements in quadratic extensions

TL;DR

This paper investigates properties of 2-primitive elements in quadratic extensions over finite fields, proving the translate and line properties hold for most cases with specific small exceptions.

Contribution

It establishes the conditions under which quadratic extensions possess translate and line properties for 2-primitive elements, identifying exact exceptions.

Findings

Translate property holds for all but six small q values.

Line property holds for all but eight small q values.

Provides theoretical and computational verification for these properties.

Abstract

Let $r,n>1$ be integers and $q$ be any prime power $q$ such that $r\mid q^n-1$. We say that the extension $\mathbb{F}_{q^n}/\mathbb{F}_q$ possesses the line property for $r$-primitive elements if, for every $\alpha,\theta\in\mathbb{F}_{q^n}^*$, such that $\mathbb{F}_{q^n}=\mathbb{F}_q(\theta)$, there exists some $x\in\mathbb{F}_q$, such that $\alpha(\theta+x)$ has multiplicative order $(q^n-1)/r$. Likewise, if, in the above definition, $\alpha$ is restricted to the value $1$, we say that $\mathbb{F}_{q^n}/\mathbb{F}_q$ possesses the translate property. In this paper we take $r=n=2$ (so that necessarily $q$ is odd) and prove that $\mathbb{F}_{q^2} /\mathbb{F}_q$ possesses the translate property for 2-primitive elements unless $q \in \{5,7,11,13,31,41\}$. With some additional theoretical and computational effort, we show also that $\mathbb{F}_{q^2} /\mathbb{F}_q$ possesses the line property for 2-primitive elements unless $q \in \{3,5,7,9,11,13,31,41\}$.