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 $K$?



Are the rationals definable in any number field?
According to R. S. Rumely, Undecidability and Definability for the theory of global fields, AMS Trans., 262, pp. 195-217, prime subfield is always definable in global field, and in number case, you can define $\Bbb N$.


