Another irreducibility criterion

TL;DR

This paper introduces a new irreducibility criterion for primitive polynomials over integers, based on bounds involving a real number and prime or prime-power conditions on polynomial evaluations.

Contribution

It provides a novel irreducibility test using inequalities and prime conditions, extending existing criteria for polynomial irreducibility.

Findings

New irreducibility criterion based on polynomial evaluation and prime conditions.

Applicable to primitive polynomials with specific coefficient bounds.

Offers a practical method for verifying irreducibility in algebraic number theory.

Abstract

Let $f=a_0+ a_{1}x+\cdots+a_m x^m\in \Bbb{Z}[x]$ be a primitive polynomial. Suppose that there exists a positive real number $\alpha$ such that $|a_m| \alpha^m>|a_0|+|a_1|\alpha+\cdots+|a_{m-1}|\alpha^{m-1}$. We prove that if there exist natural numbers $n$ and $d$ satisfying $n\geq \alpha+ d$ for which either $|f(n)|/d$ is a prime, or $|f(n)|/d$ is a prime-power coprime to $|f'(n)|$, then $f$ is irreducible in $\mathbb{Z}[x]$.