A TOPOLOGICAL APPROACH TO UNDEFINABILITY IN ALGEBRAIC EXTENSIONS OF Q

For any subset Z ⊆ Q, consider the set S_Z 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 S_Z 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 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 ∃-definable subset of an algebraic extension of Q is a finite union of single points and projections of hypersurfaces defined by absolutely irreducible polynomials.