On primitive elements in finite fields of low characteristic

TL;DR

This paper investigates the distribution of primitive elements in finite fields of low characteristic, providing new bounds on character sums that improve understanding of primitive element construction.

Contribution

It proves an unconditional bound for character sums over short intervals in function fields, advancing the theoretical understanding of primitive elements in finite fields.

Findings

Expected number of primitive elements in low degree polynomials

Unconditional bounds on character sums in function fields

Improved results over previous conditional bounds

Abstract

We discuss the problem of constructing a small subset of a finite field containing primitive elements of the field. Given a finite field, $\mathbb{F}_{q^n}$, small $q$ and large $n$, we show that the set of all low degree polynomials contains the expected number of primitive elements. The main theorem we prove is a bound for character sums over short intervals in function fields. Our result is unconditional and slightly better than what is known (conditionally under GRH) in the integer case and might be of independent interest.