The estimation of the number of Irreducible Binomials

TL;DR

This paper provides a precise estimate for counting monic irreducible binomials over finite fields with degrees up to a large threshold, enhancing understanding of polynomial structures in finite field theory.

Contribution

It introduces a sharp estimation method for the total number of monic irreducible binomials in finite fields for large degree bounds.

Findings

Derived a sharp estimate for the count of irreducible binomials.

Applicable for large degree bounds T.

Improves existing counting techniques in finite field theory.

Abstract

Let $\mathbb{F}_q$ be the finite field with $q$ elements, and $T$ a positive integer. In this article we find a sharp estimative of the total number of monic irreducible binomials in $\mathbb F_q[x]$ of degree less or equal to $T$, when $T$ is large enough.