Polynomials with factors of the form (xq –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 , … , 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 a 1 , a 2 , … , a l is a perfect qth 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 ∈ ( F 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 aj .