Vol. 9, No. 3, 2016

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The irreducibility of polynomials related to a question of Schur

Lenny Jones and Alicia Lamarche

Vol. 9 (2016), No. 3, 453–464
Abstract

In 1908, Schur raised the question of the irreducibility over $ℚ$ 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.

Keywords
irreducible polynomial
Mathematical Subject Classification 2010
Primary: 12E05, 11C08