In 2009, Jochen Koenigsmann [showed](https://annals.math.princeton.edu/wp-content/uploads/Koenigsmann.pdf) that $\mathbb{Z}$ is universally definable in the field $\mathbb{Q}$. And in 2012, Jennifer Park [proved a result](https://arxiv.org/abs/1202.6371) which implies that $\mathbb{Z}$ is $\exists\forall$-definable in many if not all finite extensions of $\mathbb{Q}$. But I’m wondering about infinite extensions.


My question is, is there a low-complexity definitions of $\mathbb{Z}$ in some infinite extension of $\mathbb{Q}$? What is the lowest-known complexity for a definition of $\mathbb{Z}$ in some infinite extension of $\mathbb{Q}$?



What is the lowest complexity definition of $\mathbb{Z}$ in an infinite extension of $\mathbb{Q}$
There is an existential definition of $\mathbb{Z}$ in the rational function field $\mathbb{R}(t)$ by a beautiful result of Denef using elliptic curves (Proposition 2 of *The diophantine problem for polynomial rings and fields of rational functions*, Trans. Amer. Math. Soc. **242** (1978), 391-399 doi:[10.1090/S0002-9947-1978-0491583-7](https://doi.org/10.1090/S0002-9947-1978-0491583-7)).


Regarding *algebraic* infinite extensions of $\mathbb{Q}$ I'm not aware of any such result. In [this](https://mathoverflow.net/questions/333028/definability-of-the-ring-of-integer-in-algebraic-extensions-of-mathbb-q) related discussion, Tom Scanlon mentions "natural infinite algebraic extensions in which $\mathbb{Z}$ is definable", but I'm not sure what he is referring to, and what the complexity of the definition would be.


