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In 2009, Jochen Koenigsmann showed that $\mathbb{Z}$ is universally definable in the field $\mathbb{Q}$. And in 2012, Jennifer Park proved a result 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 $\mathbb{Q}$?

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    $\begingroup$ It is a theorem of Fehm that a large field cannot existentially define an infinite proper subfield, so a large characteristic zero field cannot existentially define $\mathbb{Z}$. This gives a lower bound over many fields of interest $\endgroup$
    – Erik Walsberg
    Commented Feb 4, 2021 at 5:26

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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).

Regarding algebraic infinite extensions of $\mathbb{Q}$ I'm not aware of any such result. In this 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.

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