Let $K$ be a number field. Is it necessarily true that $\mathbb{Q}$ is a first-order definable subset of $K$? Equivalently (since in any number field, its ring of integers is a definable subset), is $\mathbb{Z}$ necessarily a definable subset of $K$, or of $\cal{O}_K$?

Edit: And if not, is $\mathbb{Q}$ always interpretable in $K$?

Elimination theory for the ring of algebraic integers, and Prestel and Schmid,Existentially closed domains with radical relations). $\endgroup$ – Emil Jeřábek Mar 28 '18 at 16:13