The irreducibility of polynomials related to a question of Schur

Jones, Lenny; Lamarche, Alicia

doi:10.2140/involve.2016.9.453

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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

DOI: 10.2140/involve.2016.9.453

Abstract

In 1908, Schur raised the question of the irreducibility over $ℚ$ of polynomials of the form $f (x) = (x + a_{1}) (x + a_{2}) \dots (x + a_{m}) + c$ , where the $a_{i}$ are distinct integers and $c \in {- 1, 1}$ . Since then, many authors have addressed variations and generalizations of this question. In this article, we investigate the irreducibility of $f (x)$ and $f (x^{2})$ , 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

Vol. 9, No. 3, 2016