On polynomials with roots modulo almost all primes
Volume 205 / 2022
Acta Arithmetica 205 (2022), 251-263
MSC: Primary 11R09; Secondary 11R32.
DOI: 10.4064/aa220407-9-7
Published online: 2 September 2022
Abstract
Call a monic integer polynomial exceptional if it has a root modulo all but a finite number of primes, but does not have an integer root. We classify all irreducible monic integer polynomials $h$ for which there is an irreducible monic quadratic $g$ such that the product $gh$ is exceptional. We construct exceptional polynomials with all factors of the form $X^{p}-b$ with $p$ prime and $b$ square-free.
