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In 2009, Jochen Koenigsmann showed that $\mathbb{Z}$ is universally definable in the field $\mathbb{Q}$. My question is, are there any other number fields in which $\mathbb{Z}$ is universally definable?

Or failing that, what is the lowest complexity definition of $\mathbb{Z}$ known for a number field other than $\mathbb{Q}$?

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    $\begingroup$ Of course there are the results of Jennifer Park generalizing Koenigsmann's result to number fields, but in the sense of giving a universal definition of the ring of integers of $K$ in $K$... Combining that with the results of Denef and others one can at least get an $\exists\forall$-definition of $\mathbb{Z}$ in $K$ for many $K$. $\endgroup$
    – Arno Fehm
    Commented Jan 24, 2021 at 7:20
  • $\begingroup$ @ArnoFehm I think this should be an answer. $\endgroup$
    – Emil Jeřábek
    Commented Jan 24, 2021 at 8:41
  • $\begingroup$ @EmilJeřábek: Ok, will do so, but I'm not sure that one doesn't know an $\forall$-definition at least in some cases. $\endgroup$
    – Arno Fehm
    Commented Jan 24, 2021 at 8:55

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Koenigsmann's result was generlized by Jennifer Park to number fields, giving a universal definition of the ring of integers $\mathcal{O}_K$ in $K$. Then there is a series of results proving that $\mathbb{Z}$ is existentially definable in $\mathcal{O}_K$ for certain $K$, starting I think with results by Jan Denef (totally real number fields, or quadratic extension of a totally real field). Combining the two, one gets an $\exists\forall$-definition of $\mathbb{Z}$ in $K$ at least for these number fields.

It is possible though that one also has an $\forall$-definition of $\mathbb{Z}$ in $K$ for some number fields $K\neq\mathbb{Q}$, but I don't remember right now.

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