The modified prime sieve for primitive elements in finite fields

Abstract

Let $r\ge 2$ be an integer, $q$ a prime power and ${\mathbb{F}}_q$ the finite field with $q$ elements. Consider the problem of showing existence of primitive elements in a subset $\mathcal{A}\subseteq {\mathbb{F}}_{q^r}$. We prove a sieve criterion for existence of such elements, dependent only on an estimate for the character sum ${\sum }_{{\hat{I}}^3\in \mathcal{A}}\ddot{I}‡({\hat{I}}^3)$. The flexibility and direct applicability of our criterion should be of considerable interest for problems in this field. We demonstrate the utility of our result by tackling a problem of Fernandes and Reis (2021) with $\mathcal{A}$ avoiding affine hyperplanes, obtaining significant improvements over previous knowledge.

20 pages