An irreducible class of polynomials over integers

Abstract

In this article, we consider polynomials of the form $f(x)=a_0+a_{n_1}x^{n_1}+a_{n_2}x^{n_2}+\dots+a_{n_r}x^{n_r}\in \mathbb{Z}[x],$ where $|a_0|\ge |a_{n_1}|+\dots+|a_{n_r}|,$ $|a_0|$ is a prime power and $|a_0|\nmid |a_{n_1}a_{n_r}|$. We will show that under the strict inequality these polynomials are irreducible for certain values of $n_1$. In the case of equality, apart from its cyclotomic factors, they have exactly one irreducible non-reciprocal factor.