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

Abstract

For any subset $Z\subseteq \mathbb{Q}$, consider the set $S_Z$ of subfields $L\subseteq \overline{\mathbb{Q}}$ which contain a co-infinite subset $C\subseteq L$ that is universally definable in $L$ such that $C\cap \mathbb{Q}=Z$. Placing a natural topology on the set $\text{Sub}(\overline{\mathbb{Q}})$ of subfields of $\overline{\mathbb{Q}}$, we show that if $Z$ is not thin in $\mathbb{Q}$, then $S_Z$ is meager in $\text{Sub}(\overline{\mathbb{Q}})$. Here, thin and meager both mean "small", in terms of arithmetic geometry and topology, respectively. For example, this implies that only a meager set of fields $L$ have the property that the ring of algebraic integers ${\mathcal{O}}_L$ is universally definable in $L$. The main tools are Hilbert's Irreducibility Theorem and a new normal form theorem for existential definitions. The normal form theorem, which may be of independent interest, says roughly that every $\exists$-definable subset of an algebraic extension of $\mathbb{Q}$ is a finite union of single points and projections of hypersurfaces defined by absolutely irreducible polynomials.

24 pages. Introduction has been rewritten

Cited by in corpus (2)

• Non-definability of rings of integers in most algebraic fields
• Existential rank and essential dimension of diophantine sets