Primitive elements and $k$-th powers in finite fields

TL;DR

This paper proves that for sufficiently large finite fields, there exists a primitive element such that a quadratic polynomial evaluated at this element is a $k$-th power, confirming a conjecture by Sun.

Contribution

It establishes the existence of such primitive elements in large finite fields and confirms Sun's conjecture.

Findings

Existence of primitive elements with quadratic polynomial values as $k$-th powers in large finite fields

Confirmation of Sun's conjecture on primitive elements and $k$-th powers

Explicit bounds on the size of the finite field for the result to hold

Abstract

Let $\mathbb{F}_q$ be the finite field of $q$ elements, and let $k\mid q-1$ be a positive integer. Let $f(x)=ax^2+bx+c$ be a quadratic polynomial in $\mathbb{F}_q[x]$ with $b^2-4ac\ne0$. In this paper, we show that if $q>\max\{e^{e^3},(2k)^6\}$, then there is a primitive element $g$ of $\mathbb{F}_q$ such that $f(g)\in\mathbb{F}_q^{\times k}=\{x^k: x\in\mathbb{F}_q\setminus\{0\}\}$. Moreover, we shall confirm a conjecture posed by Sun.