On polynomials with roots modulo almost all primes

TL;DR

This paper classifies certain irreducible polynomials over integers that have roots modulo almost all primes but no integer roots, and constructs specific examples with particular factorization properties.

Contribution

It provides a complete classification of irreducible monic polynomials related to exceptional polynomials and constructs explicit examples with special factorization forms.

Findings

Classified all irreducible monic integer polynomials linked to exceptional polynomials.

Constructed explicit examples with factors of the form X^p - b, where p is prime and b is square-free.

Identified conditions under which polynomials are exceptional based on their roots modulo primes.

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 $X^{p}-b$, $p$ prime and $b$ square free.