An extension of Schur's irreducibility result

Abstract

Let $n\geq 2$ be an integer. Let $\phi(x)$ 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 $\deg \phi(x)$ 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 $\sum\limits_{i=0}^{n-1} a_i(x)\frac{\phi(x)^i}{i!}+a_n\frac{\phi(x)^n}{n!}$ is irreducible over the field $\mathbb{Q}$ of rational numbers. This generalizes a well known result of Schur which states that the polynomial $\sum\limits_{i=0}^{n} a_i\frac{x^i}{i!}$ is irreducible over $\mathbb{Q}$ for all $n\geq 1$ 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 Sch\"{o}nemann Irreducibility Criterion.