## Involve: A Journal of Mathematics

• Involve
• Volume 9, Number 3 (2016), 453-464.

### The irreducibility of polynomials related to a question of Schur

#### Abstract

In 1908, Schur raised the question of the irreducibility over $ℚ$ of polynomials of the form $f(x) = (x + a1)(x + a2)⋯(x + am) + c$, where the $ai$ are distinct integers and $c ∈{−1,1}$. Since then, many authors have addressed variations and generalizations of this question. In this article, we investigate the irreducibility of $f(x)$ and $f(x2)$, where the integers $ai$ are consecutive terms of an arithmetic progression and $c$ is a nonzero integer.

#### Article information

Source
Involve, Volume 9, Number 3 (2016), 453-464.

Dates
Revised: 18 May 2015
Accepted: 17 June 2015
First available in Project Euclid: 22 November 2017

https://projecteuclid.org/euclid.involve/1511371025

Digital Object Identifier
doi:10.2140/involve.2016.9.453

Mathematical Reviews number (MathSciNet)
MR3509338

Zentralblatt MATH identifier
1342.12004

Keywords
irreducible polynomial

#### Citation

Jones, Lenny; Lamarche, Alicia. The irreducibility of polynomials related to a question of Schur. Involve 9 (2016), no. 3, 453--464. doi:10.2140/involve.2016.9.453. https://projecteuclid.org/euclid.involve/1511371025

#### References

• M. A. Bennett, “On the number of solutions of simultaneous Pell equations”, J. Reine Angew. Math. 498 (1998), 173–199.
• A. Bremner, J. H. Silverman, and N. Tzanakis, “Integral points in arithmetic progression on $\smash{y\sp 2}=x(\smash{x\sp 2}-\smash{n\sp 2})$”, J. Number Theory 80:2 (2000), 187–208.
• Y. Bugeaud, C. Levesque, and M. Waldschmidt, “Équations de Fermat–Pell–Mahler simultanées”, Publ. Math. Debrecen 79:3-4 (2011), 357–366.
• W. Flügel, “Solution to problem 226”, Archiv. der Math. und Physik 15 (1909), 271.
• K. Győry, L. Hajdu, and R. Tijdeman, “Irreducibility criteria of Schur-type and Pólya-type”, Monatsh. Math. 163:4 (2011), 415–443.
• N. Koblitz, Introduction to elliptic curves and modular forms, 2nd ed., Graduate Texts in Mathematics 97, Springer, New York, 1993.
• A. Schinzel, Selected topics on polynomials, University of Michigan Press, Ann Arbor, MI, 1982.
• A. Schinzel, Polynomials with special regard to reducibility, Encyclopedia of Mathematics and its Applications 77, Cambridge University Press, 2000.
• I. Schur, “Problem 226”, Archiv Math. Physik 13:3 (1908), 367.
• I. Seres, “Lösung und Verallgemeinerung eines Schurschen Irreduzibilitätsproblems für Polynome”, Acta Math. Acad. Sci. Hungar. 7 (1956), 151–157.
• J.-A. Serret, Cours d'algèbre supérieure, II, 4th ed., Éditions Jacques Gabay, Sceaux, 1992.
• J. H. Silverman, The arithmetic of elliptic curves, 2nd ed., Graduate Texts in Mathematics 106, Springer, Dordrecht, 2009.
• L. Weisner, “Criteria for the irreducibility of polynomials”, Bull. Amer. Math. Soc. 40:12 (1934), 864–870.
• J. Westlund, “On the irreducibility of certain polynomials”, Amer. Math. Monthly 16:4 (1909), 66–67.