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

Abstract

Given an odd prime q, a natural number l and non-zero q-free integers $a_1,a_2,\dots ,a_l$, 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 $l\le q$ and none of $a_1,a_2,\dots ,a_l$ is a perfect qth 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 ${(c_j)}_{j=1}^l\in {({\mathbb{F}}_q\setminus \{0\})}^l$, the polynomial ${\prod }_{j=1}^l(x^q-a_j)$ has roots modulo every positive integer if and only if ${\prod }_{j=1}^l(x^q-{\text{rad}}_q(a_j^{c_j})))$ has roots modulo every positive integer. Here ${\text{rad}}_q(a_j)$ denotes the q-free part of the integer a_j.

KEYWORDS:

• Power residue
• intersective polynomials

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 11A07
• 11C08
• Secondary: 11B10