Large Normal Extensions of Hilbertian Fields

Let K be a countable separably Hilbertian field. Denote the absolute Galois group of K by G(K). For each σ ∈ (σ1, . . . , σe) ∈ G(K) let Ks[σ] be the maximal Galois extension of K which is fixed by σ1, . . . , σe. We prove that for almost all σ ∈ G(K) (in the sense of the Haar measure) the field Ks[σ] is PAC and its absolute Galois group is isomorphic to F̂ω. Mathematische Zeitschrift 224 (1997), 555-565 * This research was supported by The Israel Science Foundation administered by The Israel Academy of Sciences and Humanities Introduction The goal of this note is to consider a certain natural family of closed normal subgroups of G(Q) and to prove that each group in this family is free. More generally, consider a countable separably Hilbertian field K. Denote the absolute Galois group of K by G(K). Then, for almost all σ ∈ G(K) the field Ks(σ) is PAC and e-free [FJ2, Thms. 16.13 and 16.18]. Here Ks is the separable closure of K and Ks(σ) is the fixed field of σ in Ks. Being PAC means that every nonvoid absolutely irreducible variety defined over Ks(σ) has a Ks(σ)-rational point. We say that Ks(σ) is e-free if G(Ks(σ)) (i.e., the closed subgroup 〈σ1, . . . , σe〉 of G(K) generated by σ1, . . . , σe) is free on e generators. Denote the largest Galois extension of K which is contained in Ks(σ) by Ks[σ]. It is the intersection of all K-conjugates of Ks(σ) and also the fixed field of the smallest closed normal subgroup of G(K) which contains σ1, . . . , σe. If char(K) = 0, then, for almost all σ ∈ G(K) the field Ks[σ] is PAC [FJ2, Thm. 16.47]. Lemma 1.2 below generalizes this result to arbitrary characteristic. If we knew that Ks[σ] is separably Hilbertian, then a theorem of Fried-Völklein and Pop would imply that Ks[σ] is ωfree. That is, G(Ks[σ]) is isomorphic to the free profinite group F̂w on countably many generators. Unfortunately, it is not clear how to prove the Hilbertianity of almost all Ks[σ] directly. So, we use instead a forerunner to the above mentioned theorem of Fried-Völklein-Pop and a recent theorem of Neumann [Neu] to prove directly that G(Ks[σ]) ∼= F̂ω for almost all σ ∈ G(K). A theorem of Roquette, then implies that Ks[σ] is also separably Hilbertian. Let K̃ be the algebraic closure of K. Denote the maximal purely inseparable extension of Ks[σ] by K̃[σ]. Then, for almost all σ ∈ G(K), the field K̃[σ] is PAC and ω-free. If K is a given finitely generated field, this information leads, via Galois stratification, to a primitive recursive decision procedure for the elementary theory of the family of almost all fields K̃[σ]. Acknowledgement: The author is indebted to Aharon Razon and Dan Haran for critical reading which led to improvement of an earlier version of this work.