Relatively Projective Groups as Absolute Galois Groups

Abstract

n i=1 G i is an absolute Galois group of a field of characteristic p.Mel'nikov [Mel, Thm. 1.4], proves the Theorem when rank(G i ) ≤ ℵ 0 , i = 1, . . . , n.His proof uses a theorem of Geyer [Gey]: Suppose M and L are Henselian fields with respect to rank 1 valuations and both are separable algebraic extensions of a countableHere G(K) is the absolute Galois group of K and "almost all" is used in the sense of the Haar measure of G(K). Mel'nikov's proof does not extend to the case of uncountable rank.Ershov [Ers, Thm. 3] proves part (a) of the theorem by a different method. Our proof simplifies that of Ershov. We take this opportunity to supply proofs to well known results which are not well documented in the literature.The Theorem is also a consequence of [Pop, Thm. 3.4]. As in the proof of Theorem 3.4, one may use Proposition 2.5 to obtain, in the terminology of [Pop, Ch. 3, §2], a Galois approximation of * n i=1 G i . Then one map apply [Pop, Thm. 3.4].

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

“…In the p-adic case this follows from [HJPb,Proposition 8.2(h)]. In the real casē E is real closed and our conclusion follows.…”

“…By [HJPb,Lemma 10.1], also A 2 is closed in AlgExt(E). Consequently, AlgExt(E, v) is closed in AlgExt(E).…”

“…This means that every absolutely irreducible variety V defined over K tot,S 1 (σ) with a simple K ρ v -rational point for all v and ρ has a K tot,S 1 (σ)-rational point. It follows that Gal(K tot,S 1 (σ)) is relatively projective with respect to the set {Gal(K ρ v ) | v ∈ S 1 , ρ ∈ Gal(K)} [HJPb,Proposition 4.1]. Thus, each finite embedding problem for Gal(K tot,S 1 (σ)) which has a local weak solution for each subgroup Gal(K ρ v ) has a global weak solution.…”

“…because it has a finite residue field. Nevertheless Gal(Ē) is finitely generated [HJPb,Prop. 8.2(k)].…”

“…Assume that K ′ ⊂ K ′′ . Since the absolute Galois group of a real closed field is of order 2 while the absolute Galois group of a p-adically closed field is torsion free [HJPb,Lemma 8.3], neither K ′ nor K ′′ are real closed.…”