Elements of high order in finite fields specified by binomials

TL;DR

This paper presents explicit constructions of high-order elements in finite field extensions defined by binomials, improving previous bounds on element order for certain irreducible polynomials.

Contribution

It introduces two methods to explicitly construct elements with higher multiplicative order in finite fields defined by binomials, surpassing earlier bounds.

Findings

Elements with multiplicative order at least 5^{cube root of m/2} are constructed.

The methods improve bounds on element order in certain finite field extensions.

Explicit constructions are provided for elements in fields defined by x^m - a.

Abstract

Let $F_q$ be a field with $q$ elements, where $q$ is a power of a prime number $p\geq 5$. For any integer $m\geq 2$ and $a\in F_q^*$ such that the polynomial $x^m-a$ is irreducible in $F_q[x]$, we combine two different methods to construct explicitly elements of high order in the field $F_q[x]/\langle x^m-a\rangle $. Namely, we find elements with multiplicative order of at least $5^{\sqrt[3]{m/2}}$, which is better than previously obtained bound for such family of extension fields.