The irreducibility of polynomials related to a question of Schur

Abstract

In 1908, Schur raised the question of the irreducibility over $\mathbb{Q}$ of polynomials of the form $f\left(x\right)=\left(x+a_1\right)\left(x+a_2\right)\cdots \left(x+a_m\right)+c$, where the $a_i$ are distinct integers and $c\in \left\{-1,1\right\}$. Since then, many authors have addressed variations and generalizations of this question. In this article, we investigate the irreducibility of $f\left(x\right)$ and $f\left(x^2\right)$, where the integers $a_i$ are consecutive terms of an arithmetic progression and $c$ is a nonzero integer.