A Topological Approach to Undefinability in Algebraic Extensions Of

Abstract

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. §1. Introduction. Let Sub(Q) denote the set of subfields of Q. Given a field L ∈ Sub(Q) and a set C ⊆ L, it is a question of general interest whether C is first-order definable in L using the language of rings. If so, one also wants to know how simple a defining formula can be. For example, results of Koenigsmann [11], extended by Park [16], have shown that in every number field K, the ring O K of algebraic integers is defined by a universal formula.Here we show that the usual situation is the opposite, not only for rings of integers but for any subset A ⊆ Q satisfying a rather general condition on A ∩ Q. Just as O L = O Q ∩ L, we write A L = A ∩ L. Placing a natural topology on Sub(Q), we will show that in most cases there is a comeager set of fields L ∈ Sub(Q) such that A L cannot be defined in L by any universal formula.Theorem 1.1. If A ⊆ Q is a subset for which A Q is coinfinite and not thin (as a subset of the Hilbertian field Q), then the following class is meager in