In 2009, Jochen Koenigsmann [showed](https://annals.math.princeton.edu/wp-content/uploads/Koenigsmann.pdf) 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 $\mathbb{Q}$?



Is $\mathbb{Z}$ universally definable in any number fields other than $\mathbb{Q}$?
Koenigsmann's result was generlized by [Jennifer Park](https://arxiv.org/abs/1202.6371) 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](https://doi.org/10.2307/1998133) (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.


