Primitive values of quadratic polynomials in a finite field

TL;DR

This paper proves that for sufficiently large finite fields, there exists a primitive root g such that Q(g) is also a primitive root, where Q is a quadratic polynomial with non-zero discriminant.

Contribution

It establishes the existence of primitive roots g with Q(g) also primitive in finite fields of size greater than 211, extending understanding of primitive element distributions.

Findings

Existence of such primitive roots for all q > 211

Applicable to quadratic polynomials with non-zero discriminant

Advances knowledge of primitive element properties in finite fields

Abstract

We prove that for all $q>211$, there always exists a primitive root $g$ in the finite field $\mathbb{F}_{q}$ such that $Q(g)$ is also a primitive root, where $Q(x)= ax^2 + bx + c$ is a quadratic polynomial with $a, b, c\in \mathbb{F}_{q}$ such that $b^{2} - 4ac \neq 0$.