A topological approach to undefinability in algebraic extensions of $\mathbb{Q}$

Kirsten Eisentraeger, Russell Miller, Caleb Springer, Linda Westrick

Published 2020-10-19Version 1

In this paper we investigate the algebraic extensions $K$ of $\mathbb{Q}$ in which we cannot existentially or universally define the ring of integers $\mathcal{O}_K$. A complete answer to this question would have important consequences. For example, the existence of an existential definition of $\mathbb{Z}$ in $\mathbb{Q}$ would imply that Hilbert's Tenth Problem for $\mathbb{Q}$ is undecidable, resolving one of the biggest open problems in the area. However, a conjecture of Mazur implies that the integers are not existentially definable in the rationals. Although proving that an existential definition of $\mathbb{Z}$ in $\mathbb{Q}$ does not exist appears to be out of reach right now, we show that when we consider all algebraic extensions of $\mathbb{Q}$, this is the generally expected outcome. Namely, we prove that in most algebraic extensions of the rationals, the ring of integers is not existentially definable. To make this precise, we view the set of algebraic extensions of $\mathbb{Q}$ as a topological space homeomorphic to Cantor space. In this light, the set of fields which have an existentially definable ring of integers is a meager set, i.e. is very small. On the other hand, by work of Koenigsmann and Park, it is possible to give a universal definition of the ring of integers in finite extensions of the rationals, i.e. in number fields. Still, we show that their results do not extend to most algebraic infinite extensions: the set of algebraic extensions of $\mathbb{Q}$ in which the ring of integers is universally definable is also a meager set.