Nondefinability of Rings of Integers in Most Algebraic Fields

We show that the set of algebraic extensions F of Q in which Z or the ring of integers O F are definable is meager in the set of all algebraic extensions.It is proved in Theorem 1.1 and Corollary 5.7 of Eisentraeger et al. [5] that the set of subfields F of Q in which one of Z, QnZ, O F , F nO F is existentially definable is a meager subset of the space E of all subfields E of Q, in the topology induced from 2 Q .In this short note, we explain how a stronger statement can be deduced from known results from field arithmetic (which, in particular, studies certain properties of algebraic extensions of Q) and model theory (which studies definable subsets in structures with certain properties).Recall that a field F is PAC if every geometrically irreducible F -variety has an F -rational point, !-free if every finite embedding problem for the absolute Galois group G F is solvable, and Hilbertian if A 1 .F / is not thin; that is, for every finitely many absolutely irreducible f 1 ; : : : ; f n 2 F ŒX; Y monic of degree at least 2 in Y , and 0 ¤ g 2 F ŒX, there exists x 2 F such that g.x/ ¤ 0 and f 1 f n .x;Y / has no zero in F (see Chapters 11, 27, and 12 and Section 13.5 of Fried and Jarden [7]). Proposition 1The set of subfields F of Q which are !-free and PAC is comeager in E. ProofWe claim that both the set P of PAC fields in E and the set H of Hilbertian fields in E are dense G ı -sets and therefore comeager.Since the union of two meager sets is meager, and Hilbertian PAC fields are !-free(see Jarden [9, Theorem 5.10.3]), this then implies the claim.The set P is dense in E, since for any finite extensions Q Â K Â L, Jarden's PAC Nullstellensatz (see [7, Theorem 18.6.1])gives a PAC field K Â F Â Q with