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A monic polynomial with integer coefficients is called intersective if it has no root in the rational numbers, but has a root modulo m for all positive integers m > 1. Equivalently, the polynomial has a root in each p-adic field ℚp. Using three different methods for forming these intersective polynomials, we produce an infinite family with Galois group A₄, an infinite family with Galois group D₅ and classify intersective polynomials with holomorph Galois group ℤ₂e × ℤ*₂edoi:10.14288/1.0314575 fatcat:dmvpfniawjbgpkafrc7nehpxry