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, $u \geqslant 1$, $\gcd(u, m) = 1$, $p \nmid a_{m}$ and $p^{u} > |a_{n}| + |a_{n-1}| + \cdots + |a_{m}|$. In particular, we show that the conjecture of Koley and Reddy is true.