An extension of a second irreducibility theorem of I. Schur

Abstract

Let $n\ne 8$ be a positive integer such that $n+1\ne 2^u$ for any integer $u\ge 2$. Let $\varphi (x)$ belonging to $\mathbb{Z}[x]$ be a monic polynomial which is irreducible modulo all primes less than or equal to $n+1$. Let $a_j(x)$ with $0\le j\le n-1$ belonging to $\mathbb{Z}[x]$ be polynomials having degree less than $\mathrm{deg}\varphi (x)$. Assume that the content of $(a_n{a}_0(x))$ is not divisible by any prime less than or equal to $n+1$. In this paper, we prove that the polynomial $f(x)=a_n\frac{\varphi (x{)}^n}{(n+1)!}+\sum _{j=0}^{n-1}{a}_j(x)\frac{\varphi (x{)}^j}{(j+1)!}$ is irreducible over the field $\mathbb{Q}$ of rational numbers. This generalises a well-known result of Schur which states that the polynomial $\sum _{j=0}^n{a}_j\frac{x^j}{(j+1)!}$ with $a_j\in \mathbb{Z}$ and $|a_0|=|a_n|=1$ is irreducible over $\mathbb{Q}$. We illustrate our result through examples.

arXiv admin note: substantial text overlap with arXiv:2305.04781. substantial text overlap with arXiv:2306.01767