Polynomials Consisting of Quadratic Factors with Roots Modulo Any Positive Integer

TL;DR

This paper introduces an infinite family of polynomials composed of quadratic factors that have roots modulo every positive integer but lack rational roots, highlighting interesting properties of polynomial roots in modular arithmetic.

Contribution

The paper constructs and analyzes an infinite family of polynomials with quadratic factors that have roots modulo all positive integers but no rational roots, expanding understanding of polynomial root behavior.

Findings

Polynomials with quadratic factors can have roots modulo all positive integers.

Such polynomials can lack rational roots despite their modular roots.

The family of polynomials demonstrates unique root properties in modular arithmetic.

Abstract

We give an infinite family of polynomials that have roots modulo every positive integer but fail to have rational roots. Each polynomial in this family is made up of monic quadratic factors that do not have linear term.