An extension of Schur's irreducibility result

Abstract

Let $n\ge 2$ be an integer. Let $\varphi (x)$ belonging to $\mathbb{Z}[x]$ be a monic polynomial which is irreducible modulo all primes less than or equal to $n$. Let $a_0(x),a_1(x),\dots ,a_{n-1}(x)$ belonging to $\mathbb{Z}[x]$ be polynomials each having degree less than $°\varphi (x)$ and $a_n$ be an integer. Assume that $a_n$ and the content of $a_0(x)$ are coprime with $n!$. In the present paper, we prove that the polynomial $\sum _{i=0}^{n-1}{a}_i(x)\frac{\varphi (x{)}^i}{i!}+a_n\frac{\varphi (x{)}^n}{n!}$ is irreducible over the field $\mathbb{Q}$ of rational numbers. This generalizes a well known result of Schur which states that the polynomial $\sum _{i=0}^n{a}_i\frac{x^i}{i!}$ is irreducible over $\mathbb{Q}$ for all $n\ge 1$ when each $a_i\in \mathbb{Z}$ and $|a_0|=|a_n|=1$. The present paper also extends a result of Filaseta thereby leading to a generalization of the classical Schönemann Irreducibility Criterion.