#### 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
##### Milestones
Received: 14 March 2015
Revised: 18 May 2015
Accepted: 17 June 2015
Published: 3 June 2016

Communicated by Kenneth S. Berenhaut
##### Authors
 Lenny Jones Department of Mathematics Shippensburg University Shippensburg, PA 17257-2299 United States Alicia Lamarche Department of Mathematics Shippensburg University Shippensburg, PA 17257-2299 United States