Topology of zero sets of polynomials with square discriminant

Let 𝒩 ≠ { 0 } be a fixed set of integers, closed under multiplication, closed under negation, or containing { ± 1 } . We prove that any zero of a polynomial in Z [ X ] whose coefficients lie in 𝒩 can be approximated in C to arbitrary precision by a zero of a polynomial in Z [ X ] with square discriminant whose coefficients also lie in 𝒩 . Hence the topology of the closure in C of the set of zeros of all such polynomials is insensitive to the discriminant being a square, in contrast to the Galois groups of the polynomials.