An irreducibility criterion for integer polynomials

Abstract

Let $f(x)=\sum _{i=0}^n{a}_i{x}^i$ be a polynomial with coefficients from the ring $\mathbb{Z}$ of integers satisfying either $(i)$ $0<a_0\le a_1\le \cdots \le a_{k-1}<a_k<a_{k+1}\le \cdots \le a_n$ for some $k$, $0\le k\le n-1$; or $(ii)$ $|a_n|>|a_{n-1}|+\cdots +|a_0|$ with $a_0\ne 0$. In this paper, it is proved that if $|a_n|$ or $|f(m)|$ is a prime number for some integer $m$ with $|m|\ge 2$ then the polynomial $f(x)$ is irreducible over $\mathbb{Z}$.