Elements of high order in finite fields specified by binomials

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.