A TOPOLOGICAL APPROACH TO UNDEFINABILITY IN ALGEBRAIC EXTENSIONS OF Q

Kirsten Eisenträger, Russell Miller, Caleb Springer, Linda Westrick

Research output: Contribution to journalArticlepeer-review

Abstract

For any subset Z ⊆ Q, consider the set SZ of subfields L ⊆ Q which contain a co-infinite subset C ⊆ L that is universally definable in L such that C ∩ Q = Z. Placing a natural topology on the set Sub(Q) of subfields of Q, we show that if Z is not thin in Q, then SZ is meager in Sub(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 OL 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 ∃-definable subset of an algebraic extension of Q is a finite union of single points and projections of hypersurfaces defined by absolutely irreducible polynomials.

Original languageEnglish (US)
Pages (from-to)626-655
Number of pages30
JournalBulletin of Symbolic Logic
Volume29
Issue number4
DOIs
StatePublished - Dec 29 2023

All Science Journal Classification (ASJC) codes

  • Philosophy
  • Logic

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