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- Title
A TOPOLOGICAL APPROACH TO UNDEFINABILITY IN ALGEBRAIC EXTENSIONS OF $\mathbb {Q}$.
- Authors
EISENTRÄGER, KIRSTEN; MILLER, RUSSELL; SPRINGER, CALEB; WESTRICK, LINDA
- 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 ${\operatorname {Sub}({\overline {\mathbb {Q}}})}$ of subfields of ${\overline {\mathbb {Q}}}$ , we show that if Z is not thin in ${\mathbb {Q}}$ , then $S_Z$ is meager in ${\operatorname {Sub}({\overline {\mathbb {Q}}})}$. 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.
- Subjects
TOPOLOGICAL groups; POLYNOMIALS; DEFINABILITY theory (Mathematical logic); SET theory; SUBSET selection
- Publication
Bulletin of Symbolic Logic, 2023, Vol 29, Issue 4, p626
- ISSN
1079-8986
- Publication type
Article
- DOI
10.1017/bsl.2023.37