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

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