Another irreducibility criterion

Abstract

Let $f=a_0+a_{1}x+\cdots +a_m{x}^m\in \mathbb{Z}[x]$ 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 $n\ge \alpha +d$ for which either $|f(n)|/d$ is a prime, or $|f(n)|/d$ is a prime-power coprime to $|f^{\prime }(n)|$, then $f$ is irreducible in $\mathbb{Z}[x]$.

arXiv comment5 pages