Let $n \neq 8$ be a positive integer such that $n+1 \neq 2^u$ for any integer $u\geq 2$. Let $\phi(x)$ 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 $0\leq j\leq n-1$ belonging to $\mathbb{ Z}[x]$ be polynomials having degree less than $\deg\phi(x)$. Assume that the content of $(a_na_0(x))$ is not divisible by any prime less than or equal to $n+1$. In this paper, we prove that the polynomial $f(x) = a_n\frac{ \phi(x)^n}{ (n+1)!}+ \sum\limits_{ j=0}^{ n-1}a_j(x)\frac{ \phi(x)^{ j}}{ (j+1)!}$ is irreducible over the field $\mathbb{ Q}$ of rational numbers. This generalises a well-known result of Schur which states that the polynomial $\sum\limits_{ j=0}^{ n}a_j\frac{ x^{ j}}{ (j+1)!}$ with $a_j \in \mathbb{ Z}$ and $|a_0| = |a_n| = 1$ is irreducible over $\mathbb{ Q}$. We illustrate our result through examples.
Categories: math.NT