Polynomials over Ring of Integers of Global Fields that have Roots Modulo Every Finite Indexed Subgroup

Abstract

A polynomial with coefficients in the ring of integers $\mathcal{O}_{K}$ of a global field $K$ is called intersective if it has a root modulo every finite-indexed subgroup of $\mathcal{O}_{K}$. We prove two criteria for a polynomial $f(x)\in\mathcal{O}_{K}[x]$ to be intersective. One of these criteria is in terms of the Galois group of the splitting field of the polynomial, whereas the second criterion is verifiable entirely in terms of constants which depend upon $K$ and the polynomial $f$. The proofs use the theory of global field extensions and upper bound on the least prime ideal in the Chebotarev density theorem.