An extension of Schur's irreducibility result

Abstract

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