Polynomials with factors of the form $(x^q-a)$ with roots modulo every integer

TL;DR

This paper characterizes when polynomials composed of factors of the form $(x^q - a)$ have roots modulo all positive integers, based on properties of the integers $a_j$ and their $q$-free parts, extending understanding of roots in modular arithmetic.

Contribution

It provides necessary and sufficient conditions for such polynomials to have roots modulo every positive integer, including new criteria involving $q$-free parts and powers, generalizing previous results.

Findings

Polynomials with factors $(x^q - a_j)$ lack roots modulo some integers if $a_j$ are not $q$-th powers.

Root existence depends on the $q$-free parts of the integers $a_j$ and their exponents.

The conditions unify and extend previous criteria for roots modulo all positive integers.

Abstract

Given an odd prime $q$, a natural number $l$ and non-zero $q$-free integers $a_{1}, a_{2}, \ldots, a_{l}$, none of which are equal to $1$ or $-1$, we give necessary and sufficient conditions for the polynomial $\prod_{j=1}^{l} (x^{q} - a_{j})$ to have roots modulo every positive integer. Consequently: (i) if $l \leq q$ and none of $a_{1}, a_{2}, \ldots, a_{l}$ is a perfect $q^{th}$ power, then the polynomial $\prod_{j=1}^{l} (x^{q} - a_{j})$ fails to have roots modulo some positive integer; $(ii)$ For every $l\in\mathbb{N}$, and every $(c_{j})_{j=1}^{l}\in\big(\mathbb{F}_{q}\setminus\{0\}\big)^{l}$, the polynomial $\prod_{j=1}^{l} (x^{q} - a_{j})$ has roots modulo every positive integer if and only if $\prod_{j=1}^{l} (x^{q} - \text{rad}_{q}\big(a_{j}^{c_{j}}\big)))$ has roots modulo every positive integer. Here $\text{rad}_{q}(a_{j})$ denotes the $q$-free part of the integer $a_{j}$.