Polynomial irreducibility via shifts

Lycan, Emi; Ponomarenko, Vadim

doi:10.2140/involve.2025.18.855

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Polynomial irreducibility via shifts

Emi Lycan and Vadim Ponomarenko

Vol. 18 (2025), No. 5, 855–860

DOI: 10.2140/involve.2025.18.855

Abstract

When factoring integer polynomials, it often helps to be able to tell if a polynomial is irreducible before trying (and failing) to find its factors. We examine one such irreducibility test presented by A. Bevelacqua and extend its applicability via shifts, or translations, of the polynomial. On the way there, we also encounter fixed divisors, Bunyakovsky’s conjecture, and a bound on the size of the complex roots of the polynomial.

Keywords

polynomial, integer polynomial, irreducibility

Mathematical Subject Classification

Primary: 11C08

Milestones

Received: 12 January 2024

Revised: 11 May 2024

Accepted: 3 July 2024

Published: 13 November 2025

Communicated by Scott T. Chapman

Authors

Emi Lycan
Hemet, CA United States

Vadim Ponomarenko
Department of Mathematics and Statistics San Diego State University San Diego, CA United States

Vol. 18, No. 5, 2025