A note on an irreducible class of polynomials over integers

Abstract

In this note, we prove an irreducibility criterion for the polynomial of the form $f(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_m{x}^m+p^u\in \mathbb{Z}[x]$, where $p$ is a prime number, $u⩾1$, $\gcd (u,m)=1$, $p\nmid a_m$ and $p^u>|a_n|+|a_{n-1}|+\cdots +|a_m|$. In particular, we show that the conjecture of Koley and Reddy is true.