The Absolute Galois Group of Subfields of the Field of Totally S-adic Numbers∗

For a finite set S of primes of a global field K and for σ1, . . . , σe ∈ Gal(K) we denote the field of totally S-adic numbers by Ktot,S , the fixed field of σ1, . . . , σe in Ktot,S by Ktot,S(σ), and the maximal Galois extension of K in Ktot,S(σ) by Ktot,S [σ]. We prove that for almost all σ ∈ Gal(K) the absolute Galois group of Ktot,S [σ] is isomorphic to the free product of F̂ω and a free product of local factors over S. MR Classification: 12E30 Directory: \Jarden\Diary\HJPe 23 October, 2007 * Research supported by the Minkowski Center for Geometry at Tel Aviv University, established by the Minerva Foundation. Introduction The absolute Galois group Gal(K) of a global field K is a very complicated object whose structure seems to be unattainable at the present knowledge of Galois theory. What we do understand is the structure of absolute Galois Groups of certain families of infinite extensions of K of a ”semi-local type”. The present work proves what is perhaps the ultimate word in a series of results in this subject that started forty years ago. The first major result in this direction was discovered around 1970. For σ = (σ1, . . . , σe) ∈ Gal(K) we denote the fixed field of σ1, . . . , σe in the separable closure Ks of K by Ks(σ). We denote the maximal Galois extension of K in Ks(σ) by Ks[σ]. Then, for almost all σ ∈ Gal(K) (in the sense of the Haar measure) Gal(Ks(σ)) is the free profinite group F̂e on e generators [FrJ05, Thm. 18.5.6]. The case where K = Q and e = 1 is due to James Ax [Ax67, p. 177]. In addition, for almost all σ ∈ Gal(K) Gal(Ks[σ]) is isomorphic to the free profinite group F̂ω on countably many generators [Jar97, Thm. 2.7]. On the other hand we consider a finite set S of primes of K. For each p ∈ S we choose a p-closure Kp of K at p. This is a Henselian closure if p is nonarchimedean, a real closure if p is real archimedean, and the algebraic closure K̃ of K if p is complex archimedean. Let Ktot,S = ⋂