TY - JOUR

T1 - A TOPOLOGICAL APPROACH TO UNDEFINABILITY IN ALGEBRAIC EXTENSIONS OF Q

AU - Eisenträger, Kirsten

AU - Miller, Russell

AU - Springer, Caleb

AU - Westrick, Linda

N1 - Publisher Copyright:
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Association for Symbolic Logic.

PY - 2023/12/29

Y1 - 2023/12/29

N2 - 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.

AB - 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.

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U2 - 10.1017/bsl.2023.37

DO - 10.1017/bsl.2023.37

M3 - Article

AN - SCOPUS:85173727573

SN - 1079-8986

VL - 29

SP - 626

EP - 655

JO - Bulletin of Symbolic Logic

JF - Bulletin of Symbolic Logic

IS - 4

ER -