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

2024/08/11 by Mishra, Bhawesh · 1 citation
#11A07 #11C08 Secondary 11B10 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2408.05872

Abstract

Given an odd prime q, a natural number l and non-zero q-free integers a₁, a₂, …, a_l, none of which are equal to 1 or -1, we give necessary and sufficient conditions for the polynomial ∏_j=1^l (x^q - a_j) to have roots modulo every positive integer. Consequently: (i) if l ≤ q and none of a₁, a₂, …, a_l is a perfect q^th power, then the polynomial ∏_j=1^l (x^q - a_j) fails to have roots modulo some positive integer; (ii) For every l∈ℕ, and every (c_j)_j=1^l∈(\mathbbF_q∖\0\)^l, the polynomial ∏_j=1^l (x^q - a_j) has roots modulo every positive integer if and only if ∏_j=1^l (x^q - rad_q(a_j^c_j))) has roots modulo every positive integer. Here rad_q(a_j) denotes the q-free part of the integer a_j.

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