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

Abstract

Let ${\mathbb{F}}_q$ be the finite field of $q$ elements, and let $k\mid q-1$ be a positive integer. Let $f(x)=a{x}^2+bx+c$ be a quadratic polynomial in ${\mathbb{F}}_q[x]$ with $b^2-4ac\ne 0$. 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.

9 pages