Monic integer polynomials of degree n with a root in fixed number field K have natural density vanishing as O(H^{-1} log H) for n=2 and O(H^{-1}) for n>2.
Dietmann, Probabilistic Galois theory,Bull
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On the Natural Density of Monic Integer Polynomials with Roots in a Fixed Number Field
Monic integer polynomials of degree n with a root in fixed number field K have natural density vanishing as O(H^{-1} log H) for n=2 and O(H^{-1}) for n>2.