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 $K$?

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    $\begingroup$ Variation on a theme: there's no uniform way to define $\Bbb F_p$ in $\Bbb F_{p^2}$, Chatzidakis, van den Dries, Macintyre, Definable sets over finite fields, J. Reine Angew. Math. 427 $\endgroup$
    – Denis T
    Mar 28, 2018 at 15:48
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    $\begingroup$ While this is a different question, $\mathbb Q$ and $\mathbb Z$ are not definable in $\tilde{\mathbb Q}$ and $\tilde{\mathbb Z}$. In fact, the ring $\tilde{\mathbb Z}$ is decidable (see Van den Dries, Elimination theory for the ring of algebraic integers, and Prestel and Schmid, Existentially closed domains with radical relations). $\endgroup$ Mar 28, 2018 at 16:13

1 Answer 1


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 $\Bbb N$.


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