The Absolute Galois Group of the Field of Totally S-Adic Numbers

Abstract

Abstract. For a finite set S of primes of a number field K and for σ1, . . . , σe ∈ Gal(K) we denote the field of totally S-adic numbers by Ktot,S and the fixed field of σ1, . . . , σe in Ktot,S by Ktot,S(σ). We prove that for almost all σ ∈ Gal(K) e the absolute Galois group of Ktot,S(σ) is the free product ofFe and a free product of local factors over S.

The Elementary Theory of Large Fields of Totally -Adic Numbers

“…Gal(K ) inside K S . These fields K S (σ) were studied by Jarden and Razon [25], Geyer and Jarden [18], and recently in a series of papers by Haran, Jarden, and Pop [20,21]. In particular, these authors prove that, for almost all σ ∈ Gal(K ) e , in the sense of Haar measure on the compact group Gal(K ) e , the field K S (σ) satisfies a local-global principle for rational points on varieties, and its absolute Galois group has a nice description as a free product of local factors.…”

“…The notion of group piles was introduced in [20] to enrich profinite groups with extra local data (see Remark 3.6 for some history concerning such structures). We recall this notion and extend it.…”

“…7. The rigid product can be seen as a canonical version of [20,. One could define it as a subgroup pile of the fiber product in the category of deficient group piles, which always exists.…”

“…In this section, we generalize the Cantor group piles of [20]. Recall that for a group pile G we defined the deficient reduct G • , the subgroup G = G of G, and the quotient G = G/G ; see §3.…”

“…By Lemma 6.3 there exist A and B with A • =Ã and B • =B such that (ϕ: G → A, α: B → A) is a locally solvable e-bounded (and hence self-generated) embedding problem. By[HJP09, Proposition 5.3(h)], this embedding problem has a solution, which in turn induces a solution of EP .…”