Eisenträger, Kirsten; Miller, Russell; Springer, Caleb; Westrick, Linda; (2023) A topological approach to undefinability in algebraic extensions of Q. The Bulletin of Symbolic Logic , 29 (4) pp. 626-655. 10.1017/bsl.2023.37.

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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.

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