Relatively projective groups as absolute Galois groups

Jochen Koenigsmann

Dedicated to Yuri Ershov on the occasion of his 60-th birthday

Submitted to Israel Journal of Mathematics, May 1, 2000 Abstract

By two well-known results, one of Ax, one of Lubotzky and van den Dries, a profinite group is projective iff it is isomorphic to the absolute Galois group of a pseudo-algebraically closed field. This paper gives an analogous characterization of relatively projective profinite groups as absolute Galois groups of regularly closed fields.

Contents

1 Locally split embedding problems 5

1.1 Characterizing free products via locally split embedding

problems . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Characterizing relatively projective groups . . . . . . . 9

2 Tools from valuation theory 11 2.1 Absolutely defectless fields . . . . . . . . . . . . . . . . 12 2.2 Independence . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Decomposition in finite Galois extensions . . . . . . . 14

3 Free products of decomposition groups 17

4 Proof of Theorem 1 23

5 Regularly closed fields 24 5.1 Realizing free products over regularly closed fields . . 24 5.2 The absolute Galois group of regularly closed fields . . 27

5.2.1 Solvability of embedding problems is existential 27 5.2.2 Proof of Theorem 2 . . . . . . . . . . . . . . . 35

Introduction

The absolute Galois group G_{F} of a field F is the Galois group of a separable closure F^{sep} of F over F, considered as profinite group.

The free product G_{1}⋆⋯⋆G_{n} of profinite groups G_{1},…, G_{n} is a profinite group G allowing embeddings ϵ_{i} : G_{i} → G (i = 1,…, n) such that, given any homomorphisms γ_{i} : G_{i} → H (i = 1,…, n) into a profinite group H, there is a unique homomorphism γ : G → H with γ_{i} = γ ∘ ϵ_{i} for each i.

This paper has two targets: one is to give a simplified proof of the fact that the free product of finitely many absolute Galois groups is again an absolute Galois group, and the other is to describe the absolute Galois group of multiply valued fields satisfying a local-global principle for rational points of varieties.

Theorem 1 Given fields F_{1},…, F_{n}, there is a field F of characteristic 0 with G_{F} ≅ G_{F_{1}} ⋆ ⋯ ⋆ G_{F_{n}}. Moreover, if char F_{1} = … = char F_{n} = p > 0, F can also be chosen to have characteristic p.

We call a profinite group G projective [strongly projective] relative to subgroups G_{1},…, G_{n} of G if each epimorphism π : H → → G of profinite groups which splits locally (i.e. ∀i∃ρ_{i} : G_{i} → H with π ∘ ρ_{i} = id_{G_{i}}) splits globally (i.e. ∃ρ : G → H with π ∘ ρ = id_{G} [and for each i, ρ(G_{i}) is conjugate to ρ_{i}(G_{i}) in H]).

If G is projective relative to subgroups G_{1},…, G_{n}, then G embeds into G_{1} ⋆ ⋯ ⋆ G_{n} ⋆ F, where F is some free profinite group (Proposition 1.4(5)). Since any free profinite group occurs as absolute Galois group of some field of any prescribed characteristic, and since subgroups of absolute Galois groups are absolute Galois groups, Theorem 1 immediately generalizes to

Theorem 1’ Let G be a profinite group which is projective relative to subgroups G_{1},…, G_{n} and assume that each G_{i} is an absolute Galois group. Then G is an absolute Galois group of some field of characteristic 0. Moreover, if all G_{i} can be realized over fields of the same fixed positive characteristic, then so can G.

Theorem 1, which answers Problem 18 from [J], should be attributed to Florian Pop, though he never states it: Theorem 1 is a simple consequence of Theorem 3.4 in [Po1], which even allows to generalize Theorem 1 to certain infinite free products of absolute Galois groups. The theorem was first stated and proved quite differently, in [M] in the case where the G_{i} are of countable rank. Then (without the ‘Moreover’), Ershov was the first to publish a proof ([Er3], Theorem 3), which is more in the spirit of Pop’s proof.

In all approaches, the technique for realizing free products of given absolute Galois groups is valuation theoretic: find a field F where each of the given absolute Galois groups occurs as decomposition subgroup of G_{F} w.r.t. some valuation on F, and make sure that these valuations are ‘in sufficiently general position’ to ensure that the decomposition subgroups freely generate a subgroup of G_{F}. In [M], this is achieved by ‘probabilistic’ methods: if these valuations live on a countable Hilbertian field, then, with probability 1, random conjugates of the decomposition subgroups generate a free product ([Ge], Theorem 4.1). This method, however, only works for Galois groups which are isomorphic to subgroups of absolute Galois groups of countable Hilbertian fields, i.e. for countably generated absolute Galois groups. In [Po1] and [Er3], the valuations were put in sufficiently general position by constructing a field which is also regularly closed (see below) w.r.t. finitely many valuations having the prescribed decomposition groups.

Many of the arguments in our proof can be found in [Po1] and [Er3], but our proof becomes easier for three reasons: one is that it seems unnecessary to construct a multiply valued field which is regularly closed, the second is that we work with a very handy criterion for profinite groups to be the free product of given subgroups in terms of solving ‘locally split embedding problems’ (Proposition 1.2). And, thirdly, we restrict ourselves to fields with a finite (rather than a boolean) family of valuations and, thus, avoid all the machinery needed to handle the more general situation.

Theorem 1’ generalizes Theorem 3.4 in [Po1] (for finitely many subgroups), since our notion of relative projectivity happens to coincide with Pop’s. This coincidence is based on our characterization of relatively projective group (Proposition 1.4) which provides an analogue to Gruenberg’s characterization of projective groups ([Gr], Proposition 1).

Our second target is to improve another result of Pop and Ershov on the absolute Galois group of regularly closed fields. Let us recall that an n-fold valued field (F, v_{1},…, v_{n}) (with corresponding henselisations F_{1},…, F_{n}) is regularly closed (or pseudo-closed) if it satisfies a local-global-principle for rational points on varieties, i.e. if every absolutely irreducible (affine) F - variety with a simple F_{i}-rational point for each i has an F -rational point.

We shall prove

Theorem 2 Let (F, v_{1},…, v_{n}) be a regularly closed n-fold valued field, let F_{1},…, F_{n} be henselisations of F w.r.t. v_{1},…, v_{n} resp., and assume that v_{1},…, v_{n} are independent. Then G_{F} is strongly projective relative to G_{F_{1}},…, G_{F_{n}}.

Using the variant of Theorem 1’ which - as in the proofs of Pop and Ershov

- realizes strongly projective groups over regularly closed fields (Corollary 5.2), Theorem 2 can immediately be strengthened to the following relative analogue of the Ax-Lubotzky/van den Dries characterization of projective profinite groups as absolute Galois groups of PAC-fields:

Theorem 2’ Let G be a profinite group with subgroups G_{1},…, G_{n} where each G_{i} is isomorphic to some absolute Galois group. Then G is strongly projective relative to G_{1},…, G_{n} iff G is isomorphic to the absolute Galois group of a regularly closed n-fold valued field (F, v_{1},…, v_{n}), where v_{1},…, v_{n} are independent and where, for each i, the isomorphism G ≅ G_{F} maps G_{i} onto a decomposition subgroup of G_{F} w.r.t. v_{i}.

Under the hypothesis of our Theorem 2, [Po1], Theorem 3.3 resp. [Po2], Theorem 3.2, comes to the weaker conclusion that G_{F} is projective relative to G_{F_{1}},…, G_{F_{n}}, and only under additional hypotheses (e.g. that the G_{i} be isomorphic to absolute Galois groups of real or p-adically closed fields) strong projectivity has been proved ([Po1], Theorem 1.2). Erhov shows Theorem 2 ([Er2], Thm. 3) for what he calls ‘RC^{⋆}-fields’, which are special regularly closed fields, but not all regularly closed fields are RC^{⋆}-fields. Both Pop’s and Ershov’s results, however, deal with certain infinite families of valuations, not just finite ones. We will extend our Theorem 2 to this setting later.

What made our progress on Pop’s and Ershov’s achievements possible, is the observation that solvability of finite embedding problems for absolute Galois groups is an existential first-order property in the language of fields (Observation 5.3), and that the same holds for the corresponding ‘relative embedding problems’ in the language of n-fold valued fields (Proposition 5.4). This is based on a careful analysis of decomposition subfields of finite Galois extensions which goes beyond usual ramification theory (Lemma 2.8

- 2.10) and which may be of independent interest for those working on the inverse Galois problem. For example, it turns out that solvability of (relative) embedding problems over Q is a diophantine property (Corollary 5.5).

Acknowledgement: This paper was initiated by an invitation of Moshe Jarden to the Tel Aviv University in February/March 2000. I am very greatful to him and to Dan Haran for stimulating discussions on Theorem 1 of the present paper.

1 Locally split embedding problems

Let us extract from [Po1], assertion 1.1, and from Erhsov’s analysis of ‘projective Δ^{⋆}-groups’ what seem to be the ‘correct’ notions for dealing with [strongly] relatively projective profinite groups:

Definition 1.1 Let G be a profinite group and let G_{1},…, G_{n} be subgroups of G. Then a locally split embedding problem for G w.r.t. G_{1},…, G_{n} is given by a pair of epimorphisms α : G → → B, β : A → → B, where A and B are profinite groups, and by homomorphisms β_{i} : α(G_{i}) → A with β ∘ β_{i} = id_{α(G_{i})} (i = 1,…, n). The embedding problem is called finite, if A is finite. It is called reduced, if A =< im β_{1},…, im β_{n} >.

A solution of such a locally split embedding problem is a homomorphism γ : G → A with α = β ∘γ. If γ is surjective, it is called a proper solution. A solution γ is called locally exact if γ | G_{i} = β_{i} ∘α | G_{i} for each i = 1,…, n. And a solution γ is called a locally conjugate solution if γ(G_{i}) is conjugate to im β_{i} in A for each i = 1,…, n.

1.1 Characterizing free products via locally split embedding problems

Recall that a profinite group G is the free product G = G_{1} ⋆ ⋯ ⋆ G_{n} of the subgroups G_{1},…, G_{n} ≤ G, if any given homomorphisms γ_{i} : G_{i} → H (i = 1,…, n) into a profinite group H uniquely extend to a homomorphism γ : G → H (i.e. γ | G_{i} = γ_{i} for each i). Therefore, every finite locally split embedding problem for G = G_{1} ⋆ ⋯ ⋆ G_{n} w.r.t. G_{1},…, G_{n} has a locally exact solution: take H = A and γ_{i} = β_{i} ∘ α | G_{i}. We now prove the converse:

Proposition 1.2 Let G =< G_{1},…, G_{n} > be a profinite group generated by subgroups G_{1},…, G_{n} ≤ G and assume that every finite locally split embedding problem for G w.r.t. G_{1},…, G_{n} has a locally exact solution. Then G = G_{1} ⋆ ⋯ ⋆ G_{n}.

Proof: Let G′_{1},…, G′_{n} be pairwise disjoint isomorphic copies of G_{1},…, G_{n} respectively, fix isomorphisms ρ_{i} : G_{i} → G′_{i} (i = 1,…, n), let G′ := G′_{1} ⋆ ⋯ ⋆ G′_{n} and let ρ : G′ → G be the (unique) homomorphism with ρ | G′_{i} = ρ_{i}^{−1} for i = 1,…, n. We first prove the following Claim: Given any epimorphism α : G → → B onto a finite group B and given isomorphisms ϕ_{i} : α(G_{i}) → B′_{i} (i = 1,…, n), where the B′_{i} are pairwise disjoint, there is a unique epimorphism ψ : G → → B′ := B′_{1} ⋆ ⋯ ⋆ B′_{n} such that for each i, ψ | G_{i} = ϕ_{i} ∘ α | G_{i}.

To see this, let ϕ : B′ → → B be the unique homomorphism with ϕ | B′_{i} = ϕ_{i}^{−1}. Then for any open normal subgroup N < B′ with N ≤ ker ϕ we obtain a locally split embedding problem given by α : G → → B, β : A → → B and β_{i} : α(G_{i}) → A, where A := A_{N} := B′/N with canonical projection π_{N} : B′ → → A and where β_{i} := π_{N} ∘ ϕ_{i} (i = 1,…, n).

By assumption, any such embedding problem has a locally exact solution, so there is some γ = γ_{N} : G → → A with γ | G_{i} = β_{i} ∘ α | G_{i} (i = 1,…, n). Since B′ is the inverse limit of these A_{N}, and since the corresponding γ_{N} are unique (G =< G_{1},…, G_{n} >) and compatible, the inverse limit ψ = lim_{←} γ_{N} : G → B′ exists and for each i, ψ | G_{i} = lim_{←} β_{i} ∘ α | G_{i} = ϕ_{i} ∘ α | G_{i}. As G =< G_{1},…, G_{n} >, ψ is also onto and unique, and the claim is proved.

Now associate to any open normal subgroup M < G the canonical projection α_{M} : G → → B_{M} := G/M and the projection α′_{M} : G′ → → B′_{M} := B′_{M,1} ⋆ ⋯⋆B′_{M,n} induced by the canonical projections G′_{i} →→ B′_{M,i} := G′_{i}/ρ_{i}(G_{i}∩M) (i = 1,…, n). Then, for each i, there is a (unique) isomorphism ϕ_{M,i} : α_{M}(G_{i}) → B′_{M,i} such that α′_{M} | G′_{i} = ϕ_{M,i} ∘ α_{M} ∘ ρ_{i}^{−1}. The claim thus gives a unique epimorphism ψ_{M} : G → → B′_{M} with ψ_{M} | G_{i} = ϕ_{M,i}∘α_{M} | G_{i} = α′_{M} ∘ρ_{i}.

Since G = lim_{←} α_{M}(G), we have G_{i} = lim_{←} α_{M}(G_{i}), so G′_{i} = lim_{←} α′_{M}(G′_{i}) and G′ = lim_{←} α′_{M}(G′_{i}) = lim_{←} α′_{M}(G′). By the uniqueness of the ψ_{M} we thus obtain an inverse system of commutative diagrams

\begin{matrix}G′ & →^{ρ} & G \\ α′_{M} ↓ & ↙_{ψ_{M}} & ↓ α_{M} \\ B′_{M} & → & B_{M} \end{matrix}

with ψ_{M} | G_{i} = α′_{M} ∘ ρ_{i}. Thus, the inverse limit ψ := lim_{←} ψ_{M} : G → G′ exists and ψ | G_{i} = lim_{←} α′_{M} ∘ ρ_{i} = ρ_{i} for each i. Therefore, ρ ∘ ψ = id_{G} and ρ is an isomorphism.□

For the realization of free products of absolute Galois groups as absolute Galois groups (Theorem 1), the free-product criterion of Proposition 1.2 is good enough. In some situations, however, we require a criterion which is easier to check.

Proposition 1.3 Let G =< G_{1},…, G_{n} > be a profinite group generated by subgroups G_{1},…, G_{n} ≤ G. Then G = G_{1} ⋆ ⋯ ⋆ G_{n} iff every finite reduced locally split embedding problem for G w.r.t. G_{1},…, G_{n} has a proper locally conjugate solution.

Proof: For the non-trivial direction of the proof it suffices, by Proposition 1.2 that any finite locally split embedding problem for G w.r.t. G_{1},…, G_{n} has a locally exact solution. So let A, B be finite groups, α : G → → B, β : A → → B epimorphisms and β_{i} : α(G_{i}) → A homomorphisms with β ∘β_{i} = id_{α(G_{i})} (i = 1,…, n). We have to find a homomorphism γ : G → A with γ | G_{i} = β_{i} ∘ α | G_{i}. Since B =< α(G_{1}),…, α(G_{n} >), we may assume that our embedding problem is reduced.

Let A_{1},…, A_{n} be pairwise disjoint isomorphic copies of im β_{1},…, im β_{n} respectively, fix isomorphisms π_{i} : im β_{i} → A_{i}, let A′ = A_{1} ⋆ ⋯ ⋆ A_{n} and let π : A′ → → A be the epimorphism with π | A_{i} = π_{i}^{−1} (i = 1,…, n). Then for any open normal subgroup N < A′ with N ≤ ker π we can canonically lift our given locally split embedding problem from A to A_{N} := A′/N by setting α_{N} = α : G → → B, β_{N} = β ∘ π^{N} : A_{N} → → B and β_{N,i} = π_{N} ∘ π_{i} ∘ β_{i} : α(G_{i}) → A_{N} (i = 1,…, n), where π^{N} : A_{N} → → A and π_{N} : A′ → → A_{N} are the canonical projections (so π = π^{N} ∘ π_{N}):

G
↓ α
A′ →^{π_{N}} A_{N} →^{π^{N}} A →^{β} B
⋃  ⋃  ⋃
A_{i}  ←^{π_{i}}  im β_{i} ←^{β_{i}} α(G_{i})

Note that

β_{N} ∘ β_{N,i} = β ∘ π^{N} ∘ π_{N} ∘ π_{i} ∘ β_{i} = β ∘ π ∘ π_{i} ∘ β_{i} = β ∘ β_{i} = id_{α(G_{i})},

since π ∘ π_{i} | im β_{i} = id_{im β_{i}}, and that < im β_{N,1},…, im β_{N,n} >=< π_{N}(A_{1}),…, π_{N}(A_{n}) >= A_{N}.

By assumption, each of these reduced lifted locally split embedding problems has a proper locally conjugate solution, i.e. there is an epimorphism γ_{N} : G → → A_{N} with α_{N} = β_{N} ∘ γ_{N} such that γ_{N}(G_{i}) is conjugate to im β_{N,i} = π_{N}(A_{i}) for each i. Moreover, there are only finitely many locally conjugate solutions γ_{N}, because γ_{N} is uniquely determined by the images γ_{N}(G_{i}) (i = 1,…, n): γ_{N}/G_{i}) = a_{i}^{−1}(im β_{N,i})a_{i} for some a_{i} ∈ A_{N} and β_{N,i} ∘ β_{N} | im β_{N,i} = id_{im β_{N,i}}, so for g ∈ G_{i}

a_{i}γ_{N}(g)a_{i}^{−1} = (β_{N,i} ∘ β_{N})(a_{i}γ_{N}(g)a_{i}^{−1}) = β_{N,i}(β_{N}(a_{i})α_{N}(g)β_{N}(a_{i})^{−1}).

Writing A′ = lim_{←} A_{N} with N ranging over all open normal subgroups of A′ with N ≤ ker π, and writing β′ := β ∘ π, one therefore obtains an epimorphism γ′ : G → → A′ with α = β′ ∘ γ′ such that, for each i, γ′(G_{i}) is conjugate to A_{i} in A′. Moreover, for each i, ker γ′ ∩ G_{i} = ker α ∩ G_{i}, since β′ is injective on A_{i} and hence on the conjugate γ′(G_{i}) of A_{i}, and ker α ∩ G_{i} = ker(π ∘ β_{i} ∘ α) ∩ G_{i}. Therefore, there is an isomorphism ρ_{i} : A_{i} → γ′(G_{i}) with γ′ | G_{i} = (ρ_{i} ∘ π_{i} ∘ β_{i} ∘ α) | G_{i} (i = 1,…, n).

Now let ρ : A′ → A′ be the endomorphism of A′ with ρ | A_{i} = ρ_{i}. Surjectivity then passes from γ′ to ρ, and, since A′ is finitely generated (hence small), this implies that ρ is an isomorphism.

Thus ρ^{−1} ∘γ′ : G → A′ is a homomorphism with ρ^{−1} ∘γ′ | G_{i} = π_{i} ∘β_{i} ∘α | G_{i} (i = 1,…, n), and so the induced homomorphism γ = π∘ρ^{−1}∘γ′ : G → A is the desired locally exact solution of our locally split embedding problem: for each i, γ | G_{i} = β_{i} ∘ α | G_{i}.□

1.2 Characterizing relatively projective groups

Proposition 1.4 Let G be a profinite group with subgroups G_{1},…, G_{n}. Then the following are equivalent:

(1) G is [strongly] projective relative to G_{1},…, G_{n}.

(2) Every embedding problem α : G → → B, β : A → → B with a ‘local’ solution γ_{i} : G_{i} → A (i.e. α | G_{i} = β ∘ γ_{i}) for each i has a ‘global’ solution γ : G → A (i.e. α = β ∘ γ) [and γ(G_{i}) is conjugate to γ_{i}(G_{i}) in A for each i].

(3) Every locally split embedding problem for G w.r.t. G_{1},…, G_{n} has a [locally conjugate] solution.

(4) Every finite locally split embedding problem for G w.r.t. G_{1},…, G_{n} has a [locally conjugate] solution.

Moreover, any of these conditions implies [is equivalent to]

(5) There is a free profinite group F with rk F ≤ rk G such that there is an embedding ρ : G ↪ G_{1} ⋆ ⋯ ⋆ G_{n} ⋆ F [and, for each i, ρ(G_{i}) is conjugate to the factor G_{i} in G_{1} ⋆ ⋯ ⋆ G_{n} ⋆ F].

Proof: (1) ⇒ (2): Assume (1) and let α : G → → B, β : A → → B define an embedding problem for G with local solutions γ_{i} : G_{i} → A [such that γ(G_{i}) is conjugate to γ_{i}(G_{i})] for each i. Consider the fibre product A×_{B}G := {(a, g) ∈ A × G | β(a) = α(g)} with the canonical projections π_{A} : A ×_{B} G → → A and π_{G} : A ×_{B} G → → G onto the corresponding coordinate, so α ∘ π_{G} = β ∘ π_{A}. Then π_{G} has local splittings

\begin{aligned}ρ_{i} & G_{i} & → & A ×_{B} G \\ & g & ↦ & (γ_{i}(g), g) \end{aligned}

for each i, and hence, by (1), a global splitting ρ : G → A ×_{B} G [with ρ(G_{i}) conjugate to ρ_{i}(G_{i}) in A ×_{B} G, say ρ(G_{i}) = (ρ_{i}(G_{i}))^{(a_{i},g_{i})}]. Now γ := π_{A} ∘ ρ is the solution we look for: α = α ∘ π_{G} ∘ ρ = β ∘ π_{A} ∘ ρ = β ∘ γ [and γ(G_{i}) = π_{A}(ρ(G_{i})) = π_{A}((ρ_{i}(G_{i}))^{(a_{i},g_{i})}) = γ_{i}(G_{i})^{a_{i}}] for each i.

(2) ⇒ (3): Just observe that any local splitting β_{i} : α(G_{i}) → A for a locally split embedding problem α : G → → B, β : A → → B provides a local solution γ_{i} = β_{i} ∘ α | G_{i}.

(3) ⇒ (1): Any epimorphism π : H → → G with local splittings ρ_{i} : G_{i} → H defines a locally split embedding problem α = id_{G}, β = π and β_{i} = ρ_{i}.

(3) ⇒ (4): Clear.

(4) ⇒ (1): Assume (4) and let π : H → → G be an epimorphism with local splittings ρ_{i} : G_{i} → H (i = 1,…, n).

We first proceed as the proof of [Gr], Proposition 1, and consider the case where ker π is finite. For each i, ker π ∩im ρ_{i} = 1 since π ∘ρ_{i} = id_{G_{i}}. As ker π is finite, there is an open normal subgroup N < H with ker π ∩ Nim ρ_{i} = 1 for all i = 1,…, n. Hence N ker π ∩ Nim ρ_{i} = N and so the induced finite embedding problem α_{N} : G → → G/π(N), (β_{N} =) π_{N} : H/N → → G/π(N) where α_{N} is the canonical projection and π_{N}(hN) = π(h)π(N) for h ∈ H, is locally split:

ker π_{N} ∩ Nim ρ_{i}/N = (N ker π ∩ Nim ρ_{i})/N = N/N = 1.

Thus, there is a solution γ_{N} : G → H/N with π_{N} ∘ γ_{N} = α_{N}.

Now consider the fibre product

H/N ×_{G/π(N)} G := {(hN, g) ∈ H/N × G | π_{N}(hN) = α_{N}(g)}

and observe that

\begin{aligned}ϵ : & H & → & H/N ×_{G/π(N)} G \\ & h & ↦ & (hN, π(h)) \end{aligned}

is an embedding, since ker π ∩ N = 1. Now

\begin{aligned}ρ′ : & G & → & H/N ×_{G/π(N)} G \\ & g & ↦ & (γ_{N}(g), g) \end{aligned}

is well-defined (π_{N}(γ_{N}(g)) = α_{N}(g)) and im ρ′ ⊆ im ϵ: for g ∈ G let γ_{N}(g) = hN for some h ∈ H and observe that h can be chosen with π(h) = g because π(h) · π(N) = g · π(N). Hence ρ := ϵ^{−1} ∘ ρ′ is a splitting for π.

[Moreover, if γ_{N} is a locally conjugate solution, say γ_{N}(G_{i}) = (Nimρ_{i}(G_{i})/N)^{h_{i}N} for some h_{i} ∈ H, then ρ′(G_{i}) = (ϵ(Nim ρ_{i}(G_{i})/N)^{h_{i}N} and so ρ(G_{i}) is conjugate to im ρ_{i} in H.]

Further, let us observe that each [locally conjugate] splitting ρ of π can be considered as inverse limit of [l.c.] splittings ρ_{N} of π_{N}, where N runs through all open normal subgroups of H with ker π ∩ Nimρ = 1 (again, such N’s exist because ker π is finite and ker π ∩ ρ(G) = 1). Conversely, any such inverse limit of [l.c.] compatible splittings of π_{N} gives a splitting of π. Since each π_{N} has only finitely many splittings the set of [l.c.] splittings of π is an inverse limit of finite sets, and hence compact.

Now let ker π be arbitrary and consider the family K of normal subgroups K < H which are open subgroups of ker π. Then for each K ∈ K, π induces a projection π_{K} : H/K → → G which splits locally and has a finite kernel, so the set R_{K} of [locally conjugate] splittings of π_{K} is non-empty and compact. Now the R_{K} (K ∈ K) form an inverse system of non-empty compact sets. Hence the inverse limit is non-empty, and any element in it defines a [locally conjugate] splitting of π

(1) ⇒ (5): Assuming (1), we can choose a free profinite group F of rk F = rk G with an epimorphism π_{0} : F → → G extending to an epimorphism π : G_{1} ⋆ ⋯ ⋆ G_{n} ⋆ F → G which maps each free factor G_{i} identically onto the subgroup G_{i} of G. By (1), π splits globally [in a locally conjugate way] and any such splitting gives the desired embedding.

[5] ⇒ [1]: Assume [5]. Then, by the subgroup theorem of Haran ([Ha], Theorem 5.1), ρ(G) is strongly projective relative to ρ(G_{1}),…, ρ(G_{n}), since G_{1} ⋆ ⋯ ⋆ G_{n} ⋆ F is strongly projective relative to G_{1},…, G_{n}. (Compare the following remark).□

Remark 1.5 As the proposition shows, our notions of relative [strong] projectivity coincide with Pop’s notions (defined via finite embedding problems). Our notion of strong relative projectivity coincides with Haran’s notion of relative projectivity and with Ershov’s notion of a projective Δ-group: if G is strongly projective relative to subgroups G_{1},…, G_{n} then G_{1},…, G_{n} are separated, i.e. G_{i} ∩ G_{j} = 1 for i ≠ j. This follows from (2): considering the homomorphism G_{1}⋆⋯⋆G_{n}⋆F → G_{1}×⋯×G_{n}×F that identifies the factors it is clear that for i ≠ j any conjugates of G_{i} and G_{j} in G_{1} ⋆ ⋯ ⋆ G_{n} ⋆ F intersect trivially. This also answers the question implicit in [Er2], Remark 1.

2 Tools from valuation theory

In this section we describe valuation theoretic tools used to realize or to recognize absolute Galois groups as free or projective products of decomposition groups, at the same time introducing (mostly standard) notation and terminology as well as collecting other (mostly well-known) facts. [En] and [Ri] are classical references on valuation theory, the most comprehensive recent book is [K].

2.1 Absolutely defectless fields

For a valued field (F, v) we denote valuation ring, maximal ideal, residue field and value group by O_{v}, M_{v}, F v := O_{v}/M_{v} and Γ_{v} = v(F^{×}) ≅ F^{×}/O_{v}^{×} respectively. Let D_{v} be a decomposition subgroup of G_{F} w.r.t. v, i.e. D_{v} = G_{F^{v}} for some henselisation F^{v} of (F, v) in F^{sep}. Denoting ramification and inertia subgroup of D_{v} by R_{v} and I_{v}, we recall the following

Facts 2.1 (a) R_{v} and I_{v} are normal subgroups of D_{v} with R_{v} ≤ I_{v}, and both R_{v} and I_{v} have complements in D_{v} ([KPR]).

(b) D_{v}/I_{v} ≅ G_{F v}

(c) R_{v} = 1 if char F v = 0; otherwise (if char F v = p > 0), R_{v} is a Sylow-p subgroup of I_{v}.

(d) The fixed field of R_{v} in F^{sep} is the smallest subextension of F^{sep}/F^{v} with separably closed residue field and q-divisible value group for all primes q ≠ char F v.

Definition 2.2 We call a valued field (F, v) absolutely defectless if R_{v} = 1.

This definition does not depend on the choice of D_{v}, since any two decomposition subgroups of G_{F} are conjugate in G_{F} and this conjugation induces an isomorphism of the corresponding ramification subgroups. Note also, that all finite separable extensions of an absolutely defectless valued field are defectless (i.e. the fundamental equality ‘∑ e_{i} · f_{i} = n’ holds), but the converse may be false (e.g. Q_{p}). Moreover, an absolutely defectless field may well have inseparable extensions with defect. (Using the terminology of [K], chapter II.12, (F, v) is absolutely defectless iff the henselisation of (F, v) is ‘separably tame’.)

Recall that an extension (F′, v′)/(F, v) of valued fields is called immediate if the canonical embeddings F v ↪ F′v′ and Γ_{v} ↪ Γ_{v′} are isomorphisms. An extension (F′, v′_{1},…, v′_{n})/(F, v_{1},…, v_{n}) of n-fold valued fields is called immediate, if each (F′, v′_{i})/(F, v_{i}) is immediate.

Observation 2.3 Let (F′, v′)/(F, v) be an immediate extension of valued fields with (F′, v′) henselian and absolutely defectless and with F relatively algebraically closed in F′. Then (F, v) is also henselian and absolutely defectless and res : G_{F′} → G_{F} is an isomorphism.

Proof: Since F is relatively algebraically closed in F′, it is clear that (F, v) is henselian and res : G_{F′} → G_{F} is surjective. If F_{R} denotes the fixed field of the ramification subgroup R_{v} of G_{F} in F^{sep}, then, since (F′, v′)/(F, v) is immediate, F′F_{R} is an algebraic extension of (F′, v′) with separably closed residue field and q-divisible value group for all primes q ≠ char F v. Since (F′, v′) is absolutely defectless, this implies that F′F_{R} = F^{′sep} (Fact 2.1(d)), so F^{′sep} = F′F^{sep} and res : G_{F′} → G_{F} is injective. Thus, res is an isomorphism and, in particular, R_{v} = G_{F_{R}} ≅ G_{F′F_{R}} = 1, i.e. (F, v) is also absolutely defectless.□

That (F, v) in the observation is henselian and absolutely defectless was already observed in [K], II., Lemma 12.29 under the weaker assumption that F′v′/F v be algebraic (instead of (F′, v′)/(F, v) being immediate).

Caveat Note that it may happen that (F′, v′) is henselian and absolutely defectless, but that a relatively algebraically closed subfield (F, v) is not absolutely defectless.

2.2 Independence

Two valuations v and w on a field F are called independent if they are non-trivial and F = O_{v}O_{w}, i.e., as a ring, F is generated by the proper subrings O_{v} and O_{w}. An important consequence is the well-known Approximation Theorem Let v_{1},…, v_{n} be (pairwise) independent valuations on a field F. Then, given any a_{1},…, a_{n} ∈ F and b_{1},…, b_{n} ∈ F^{×}, there is an element x ∈ F with v_{i}(x − a_{i}) > v_{i}(b_{i}) for all i = 1,… n.

We shall need the following almost trivial

Observation 2.4 Two valuations v and w on a field F are independent iff ∀γ ∈ Γ_{v} ∃x ∈ F : v(x) > γ & w(x) < 0.

Proof: If v and w are independent then they are non-trivial. So, given γ ∈ Γ_{v}, there are y, z ∈ F with v(y) > γ and w(z) < 0. The approximation theorem now provides an element x ∈ F with v(x) = v(y) > γ and w(x) = w(z) < 0.

If, conversely, given any y ∈ F, we find an element x ∈ F with v(x) > v(y^{−1}) and w(x) < 0, we see that y = (yx)x^{−1} ∈ O_{v}O_{w}, and that v and w are non-trivial.□

Corollary 2.5 Let v_{1},…, v_{n} be independent valuations on F and let (F′, v′_{1},…, v′_{n})/(F, v_{1},…, v_{n}) be an extension of n-fold valued fields where each Γ_{v_{i}} is cofinal in Γ_{v′_{i}}, i.e. ∀γ ∈ Γ_{v′_{i}} ∃δ ∈ Γ_{v_{i}} with δ ≥ γ (e.g. if the extension is immediate or algebraic). Then v′_{1},…, v′_{n} are independent.

For the convenience of the reader, let us reproduce the proof of the following

Fact 2.6 ([H], Theorem 1.1) If F = F_{1} ∩ … ∩ F_{n} for henselian algebraic extensions F_{1},…, F_{n} of a field F inducing independent valuations v_{1},…, v_{n} on F, then each F_{i} is a henselisation of (F, v_{i}).

Proof: Let F^{(i)} be a henselisation of (F, v_{i}) in F_{i} and pick any α ∈ F_{i}. Then we can approximate the irreducible polynomial of α over F w.r.t. v_{i} and, for j ≠ i some polynomial of the same degree splitting in distinct linear factors over F w.r.t. v_{j} sufficiently well by some f ∈ F [X] to guarantee that all zeros of f lie in ⋂_{j≠i} F_{j} and that for some zero β of f (close to α in F_{i}) F^{(i)}(β) = F^{(i)}(α) ⊆ F_{i} (Krasner’s Lemma). But then β ∈ ⋂_{j} F_{j} = F and α ∈ F^{(i)}(α) = F^{(i)}(β) = F^{(i)}. Hence F_{i} = F^{(i)}. □

2.3 Decomposition in finite Galois extensions

Recall the following well-known details from ramification theory:

Facts 2.7 Let (F, v) be a valued field, let L/F be a finite Galois extension and let K/F be a subextension of L/F.

(i) K is a decomposition subfield of L/F w.r.t v iff K = F ∩ H for some henselisation H of (F, v). (ii) If K is a decomposition subfield of L/F w.r.t. v, say D := Gal(L/K) = {σ ∈ Gal(L/F) | σO_{w} = O_{w}} for some prolongation w of v to L, then w is the only prolongation of w | K to L and v has exactly r := [K : F] = [Gal(L/F) : D] distinct prolongations to L: the ‘conjugates’ w ∘ τ_{1},…, w ∘ τ_{r} of w, where Gal(L/F) = ⋃_{i=1}^{r} τ_{i}D. (iii) If E/F is any finite extension with distinct prolongations w_{1},…, w_{k} of v to E, then, given x_{1} ∈ O_{w_{1}},…, x_{k} ∈ O_{w_{k}}, there is some x ∈ E with x − x_{i} ∈ M_{w_{i}} for each i.

Lemma 2.8 Given a finite Galois extension L_{0}/F_{0} and an embedding ϕ : Gal(L_{0}/F_{0}) ↪ A into a finite group A, there is a Galois extension (L, w)/(F, v) of valued fields with Lw ≅ L_{0}, F v ≅ F_{0}, and Gal(L/F) = A such that ϕ(Gal(L_{0}/F_{0})) is the decomposition subgroup of A w.r.t. w. ((L, w)/(F, v) is then tame unramified, i.e. the inertia subgroup of Gal(L/F) w.r.t. w is trivial.)

Proof: Let L = F_{0}(T_{1},…, T_{d}), where d := ♯A, say A = {1 = a_{1}, a_{2},…, a_{d}} with ϕ(Gal(L_{0}/F_{0})) = {a_{1},…, a_{f}} (so f | d), and where T_{1},… T_{d} are indeterminates over F_{0}.

Let A act on L by acting trivially on F_{0}, and via ‘left multiplication’ on {T_{1},…, T_{d}}, i.e. a_{k}(T_{l}) = T_{m} ⇔ a_{k} · a_{l} = a_{m}. Denoting the fixed field under this action of A on L by F, it is clear that L/F is a Galois extension with Gal(L/F) = A.

Now choose x ∈ L_{0} such that {σ(x) | σ ∈ Gal(L_{0}/F_{0})} is a normal base for L_{0}/F_{0}. Define a ring homomorphism

\begin{aligned}π_{0} : & F_{0}[T_{1},…, T_{d}] & → & L_{0} \\ & T_{i} & ↦ & \{ \begin{matrix}ϕ^{−1}(a_{i})(x) & for i ≤ f \\ 0 & for i > f \end{matrix} \end{aligned}

with π_{0} | F_{0} = id_{F_{0}} and extend it to a place π : L → L_{0} ∪ {∞} (Chevalley-Lang) with corresponding valuation w. Then Lw ≅ L_{0} and Kw ≅ F v ≅ F_{0}, where K is the fixed field of ϕ(Gal(L_{0}/F_{0})) in L and v := w | F. Moreover, for i > f, a_{i}(O_{w}) ⊈ O_{w} (e.g. T_{1}^{−1} ∈ O_{w}, but a_{i}(T_{1}^{−1}) = T_{i} ∉ O_{w}). Hence ϕ(Gal(L_{0}/F_{0})) is the decomposition subgroup of A w.r.t. w.□

The lemma, of course, implies that any finite group A with any subgroup D ≤ A can be realized as Galois group of a Galois extension of valued fields, where D becomes a decomposition subgroup. And it is not difficult to see that this generalizes to profinite groups. It may, however, be worth noting that this has no analogue for absolute Galois groups: in general, not any subgroup D of an absolute Galois group A = G_{F} can become a decomposition subgroup of A when A is suitably realized as absolute Galois group of some valued field: e.g. if 2 < [A : D] < ∞ this is not possible.

Lemma 2.9 Let (L, w)/(F, v) be a tame unramified finite Galois extension of valued fields, where F v is infinite. Then there is a primitive element x ∈ L = F (x) over F with irreducible polynomial f ∈ O_{v}[X] over F such that \overline{x} ∈ Lw is a primitive element for Lw/F v and such that \overline{f} ∈ F v[X] is the product of the irreducible polynomials of pairwise non-conjugate primitive elements for the Galois extension Lw/F v over F v. In particular, f′(x) ∈ O_{w}^{×}.

Proof: Let K be the decompositin subfield of L/F w.r.t. w, let r := [K : F], let τ_{1} = 1, τ_{2},…, τ_{r} ∈ Gal(L/F) be representatives for the cosets of Gal(L/K) in Gal(L/F) and let w_{1} := w∘τ_{1} = w, w_{2} := w∘τ_{2},…, w_{r} := w∘τ_{r} be the distinct prolongations of v to L.

Choose x_{1} ∈ O_{w} such that \overline{x_{1}} ∈ Lw is a primitive element for Lw = F v(\overline{x_{1}}) over F v, and choose a_{1} := 1, a_{2},…, a_{r} ∈ O_{v}^{×} with \overline{a_{i}} ≠ \overline{a_{j}} for i ≠ j (this is possible as F v is infinite). Then \overline{a_{1}x_{1}},… \overline{a_{r}x_{1}} are non-conjugate primitive elements for Lw/F v.

Now choose x ∈ L with x − τ_{i}(a_{i}x_{1}) ∈ M_{w_{i}} for i = 1,…, r: use Fact 2.7(iii). Since \overline{τ_{1}^{−1}(x)} = \overline{a_{1}x_{1}},…, \overline{τ_{r}^{−1}(x)} = \overline{a_{r}x_{r}} are non-conjugate primitive elements for Lw/F v, the conjugates στ_{i}^{−1}(x) of x in L over F with σ ∈ Gal(L/K) ≅ Gal(Lw/F v) and i = 1,…, r are all distinct, so L = L(x) (as [L : F] = r · [L : K]). Moreover, the irreducible polynomial f of x over F is in O_{v}[X], and \overline{f} decomposes over F v into the product of the irreducible polynomials of \overline{a_{i}x_{1}} (i = 1,…, r) over F v, which are pairwise coprime, whence \overline{f′(x)} ≠ 0 ∈ F v.□

Lemma 2.10 Let (F, v) be a non-trivially valued field, let L/F be a finite Galois extension, let K/F be a subextension of L/F of degree [K : F] = r, and assume that v has at most r prolongations to L.

Then K is a decomposition subfield of L/F iff there is a polynomial h(X) = X^{r} + h_{r−1}X^{r−1} + … + h_{0} ∈ F [X] which is irreducible over F, has a root in K and coefficients with h_{0},… h_{r−2}, 1 + h_{r−1} ∈ M_{v}.

Proof: ‘⇐’: If h ∈ F [X] has all the properties mentioned, then K = F (x) for some root x of h in K, and, by Hensel’s Lemma, x is in some henselisation of F w.r.t. v (v(h(1)) > 0 = v′(h(1))), i.e. K is contained in a decomposition subfield E of L/F w.r.t v (Fact 2.7(i)). By Fact 2.7(ii), v has [E : F] prolongations to L, so r ≥ [E : F] ≥ [K : F] = r and hence K = E.

‘⇒’: Let K be a decomposition subfield of L/K, say w.r.t. the prolongation w of v to L, and let Gal(L/F) = ⋃_{i=1}^{r} τ_{i}Gal(L/K) with τ_{1} = 1. Then, by Fact 2.7(ii) and (iii), there is some y ∈ K with y ∈ (1+M_{w})∩⋂_{i=2}^{r} M_{w∘τ_{i}}: note that w ∘ τ_{i} | K ≠ w ∘ τ_{j} | K for i ≠ j, since w is the only prolongation of w | K to L, and so w | K ≠ w ∘ τ_{j}τ_{i}^{−1} | K.

Now let z be a primitive element for the extension K/F (y), so K = F (y, z). Since v is non-trivial and Γ_{v} is cofinal in all Γ_{w∘τ_{i}}, we find infinitely many a ∈ M_{v} ⊆ F such that az ∈ ⋂_{i=1}^{r} M_{w∘τ_{i}}. As K/F is finite, there are two such a’s, say a_{1} ≠ a_{2} ∈ F, with F (a_{1}z+y) = F (a_{2}z+y), so y, z ∈ F (a_{1}z+ y), and hence x := a_{1}z + y is a primitive element for K = F (y, z) = F (x) over F. Moreover, x ∈ (1 + M_{w}) ∩ ⋂_{i=2}^{r} M_{w∘τ_{i}}, so τ_{2}(x),…, τ_{r}(x) ∈ M_{w}, and the irreducible polynomial of x over F:

h(X) = ∏_{i=1}^{r}(X − τ_{i}(x)) ∈ F [X]

has coefficients h_{0},…, h_{r−2} ∈ M_{v} and h_{r−1} ∈ −1 + M_{v}.□

3 Free products of decomposition groups

The goal of this section is to prove the following proposition and a variation of it (Proposition 3.7).

Proposition 3.1 Let F be a field with absolutely defectless valuations v_{1},…, v_{n} and assume that for each i, (F, v_{i}) admits an immediate extension (F̂_{i}, v̂_{i})/(F, v_{i}) with ♯F̂_{i} > ♯F. Let D_{1},…, D_{n} be decomposition subgroups of G_{F} w.r.t. v_{1},…, v_{n} respectively.

Then there is an immediate extension (F′, v′_{1},…, v′_{n})/(F, v_{1} …, v_{n}) with ♯F′ = ♯F such that G_{F′} = D′_{1}⋆⋯⋆D′_{n}, where for each i, D′_{i} is a decomposition subgroup of G_{F′} w.r.t. v′_{i} and res : D′_{i} → D_{i} is an isomorphism.

Remark 3.2 In the hypothesis of the above proposition, we may even assume that for each i, (F̂_{i}, v̂_{i}) is henselian, is absolutely defectless and contains the henselisation F_{i} of (F, v_{i}) corresponding to D_{i} = G_{F_{i}}. Moreover, we may assume that G_{F} =< D_{1},…, D_{n} >, i.e. F = F_{1} ∩… ∩ F_{n}.

Proof of the remark: First pass to a henselisation (F̂′_{i}, v̂′_{i}) of (F̂_{i}, v̂_{i}). This is an immediate extension and F̂′_{i} ∩ F^{sep} is a henselisation of (F, v_{i}): since (F, v_{i}) is absolutely defectless any immediate separably algebraic henselian extension is a henselisation. So there is an isomorphism F̂′_{i} ∩ F^{sep} → F_{i} over (F, v_{i}) which extends to an isomorphism (F̂′_{i}, v̂′_{i}) → (F̂′′_{i}, v̂′′_{i}) over (F, v_{i}). Thus, (F̂′′_{i}, v̂′′_{i})/(F, v_{i}) is an immediate henselian extension containing F_{i}.

Now we pass to the fixed field F̂′′′_{i} of a complement of the ramification subgroup of GF̂′′_{i} (cf. Fact 2.1(a)) to obtain an absolutely defectless field. (F̂′′′_{i}, v̂′′′_{i})/(F, v_{i}) is still immediate since passing from F̂′′_{i} to F̂′′′_{i} means passing to the perfect hull for the residue field and passing to the p-divisible hull for the value group (p = char F v, cf. Fact 2.1(d)). As (F, v_{i}) is absolutely defectless, F v ≅ F̂′′_{i} is already perfect and Γ_{v_{i}} ≅ Γ_{}_{v̂′′_{i}} p-divisible.

Finally, replacing F by F_{1}∩…∩F_{n} doesn’t change any of the hypotheses: note that the assumption that (F̂_{i}, v̂_{i})/(F, v_{i}) be an immediate extension of higher cardinality implies that the valuations v_{1},…, v_{n} are all non-trivial, so all fields involved are infinite and hence, all algebraic extensions of the same cardinality.□

Before proving the proposition let us first isolate the key arguments in three lemmas.

Lemma 3.3 Given an n-fold valued field (F, v_{1},…, v_{n}) and decomposition groups D_{1},…, D_{n} as in Proposition 3.1 and the remark thereafter, there is an immediate extension (F′, v′_{1},…, v′_{n})/(F, v_{1} …, v_{n}) with ♯F′ = ♯F such that v′_{1},…, v′_{n} are independent, absolutely defectless, and G_{F′} =< D′_{1} …, D′_{n} > for decomposition subgroups D′_{i} of G_{F′} w.r.t. v′_{i} for which res : D′_{i} → D_{i} is an isomorphism (i = 1,…, n).

Proof: Let X = X_{1} ˙∪… ˙∪X_{n} be a (partitioned) set of indeterminates over F with ♯X_{1} = ⋯ = ♯X_{n} = ♯F. For each i, (F̂_{i}, v̂_{i})/(F, v_{i}) is immediate and ♯F̂_{i} > ♯F, so we find an embedding ϕ_{i} : F (X) → F̂_{i} such that v̂_{i}(ϕ_{i}(x)) < 0 for all x ∈ X_{j} (j ≠ i) and such that ∀γ ∈ Γ_{v_{i}}∃x ∈ X_{i} with v̂_{i}(ϕ_{i}(x)) > γ.

After passing to isomorphic copies of (F̂_{i}, v̂_{i}) over (F_{i}, v_{i}) (as in the proof of the previous remark), we may for all i = 1,…, n, x ∈ X identify ϕ_{i}(x) with x, so that then X ⊆ ⋂_{i} F̂_{i}, and for each i, v̂_{i}(X_{j}) < 0 when j ≠ i and v̂_{i}(X_{i}) is cofinal in Γv̂_{i}. Hence, by Observation 2.4, the valuations on F (X) induced by v̂_{1},…, v̂_{n} are independent.

Passing, if necessary, once more to isomorphic copies of (F̂_{i}, v̂_{i}) over (F_{i}(X), v̂_{i} | F_{i}(X)), we may even assume that the relative algebraic closures F′_{i} of F_{i}(X) in F̂_{i} are all contained in a fixed algebraic closure of F (X). By Observation 2.3, each F′_{i} is then henselian and absolutely defectless (w.r.t. v̂_{i} | F′_{i}).

Now let F′ = F′_{1} ∩ … ∩ F′_{n} and, for each i, let v′_{i} := v̂_{i} | F′. Then, by Corollary 2.5, v′_{1},… v′_{n} are independent, and so, by Fact 2.6, each F′_{i} is a henselisation of (F′, v′_{i}). Hence, by Observation 2.3, res : D′_{i} := G_{F′_{i}} → D_{i} is an isomorphism for each i: F_{i} is relatively algebraically closed in F̂_{i}, and so in F′_{i}.□

Lemma 3.4 ([HJ], Proof of Prop. 14.1, Part C, [P1], Thm. 3.1, and [Er2], Lemma 1) Let L/K be a finite Galois extension with Galois group B = Gal(L/K) and let β : A → → B be an epimorphism of finite groups. Consider the elements of A as indeterminates over L and let A act on L(A) via group multiplication on A and via the given Galois action of β(A) = B on L. Assume that D ≤ A is a subgroup with D ∩ ker β = 1.

Then L(A)^{D} := {x ∈ L(A) | σx = x for all σ ∈ D} is purely transcendental over L^{β(D)}.

Lemma 3.5 Let (F, v_{1},…, v_{n}) and D_{1},…, D_{n} satisfy the assumptions of Proposition 3.1 and the remark thereafter. Assume, in addition, that either v_{1},…, v_{n} are independent or that char F v_{i} = 0 for all i. Let α : G_{F} → → B, β : A → → B and β_{i} : α(D_{i}) → A (i = 1,…, n) be the data for a locally split embedding problem for G_{F} w.r.t. D_{1},… D_{n}, i.e. A, B are finite groups, α, β are epimorphisms, β_{1},…, β_{n} are homomorphisms with A =< im β_{1},…, im β_{n} > and, for each i, β ∘ β_{i} = id_{α(D_{i})}.

Then (F, v_{1},…, v_{n}) admits an immediate extension (F′, v′_{1},…, v′_{n}) with ♯F′ = ♯F, where each (F′, v′_{i}) is absolutely defectless and where G_{F′} has decomposition subgroups D′_{1},…, D′_{n} w.r.t. v′_{1},…, v′_{n} respectively such that G_{F′} =< D′_{1},… D′_{n} >, res : D′_{i} → D_{i} is an isomorphism for each i and such that the lifted locally split embedding problem for G_{F′} w.r.t. D′_{1},…, D′_{n} (i.e. α′ := α ∘ res : G_{F′} → → B, β′ := β : A → → B, β′_{i} := β_{i} ∘ res : α′(D′_{i}) → A) has a locally exact solution (i.e. ∃γ′ : G_{F′} → → A with γ′ | D′_{i} = β′_{i} ∘ α′ | D′_{i}).

Proof: Let L = (F^{sep})^{ker α} be the fixed field of ker α in F^{sep}, so L/F is a Galois extension with Gal(L/F) ≅ B and (identifying those two groups) we may assume that α = res : G_{F} → Gal(L/F) = B. Then, for each i, L^{α(D_{i})} = F_{i} ∩ L.

Consider the field L(A) with the A-action from the previous lemma. Then L(A)/L(A)^{A} is a Galois extension with Galois group A and, for each i, the conclusion of the lemma (with D = im β_{i}) says that the extension L(A)^{im β_{i}}/L^{α(D_{i})} is purely transcendental.

Since L^{α(D_{i})} ⊆ F̂_{i} and ♯F̂_{i} > ♯F = ♯L^{α(D_{i})}, we can consider L(A)^{im β_{i}} as subfield of F̂_{i}.

As before, passing, if necessary, to isomorphic copies of F̂_{i} over F_{i}L(A)^{im β_{i}}, we may even assume that all relative algebraic closures F′_{i} of F_{i}L(A)^{im β_{i}} in F̂_{i} are contained in a fixed algebraic closure of L(A). Then, by Observation 2.3, each F′_{i} is henselian and absolutely defectless w.r.t. v̂_{i} | F′_{i}, res : D′_{i} := G_{F′_{i}} → D_{i} is an isomorphism, and (F′ := F′_{1} ∩… ∩ F′_{n}, v′_{1},…, v′_{n}) with v′_{i} := v̂_{i} | F′ is an immediate extension of (F, v_{1},…, v_{n}) with G_{F′} =< D′_{1},…, D′_{n} > and ♯F′ = ♯F.

Moreover, each F′_{i} is a henselisation of (F′, v′_{i}) (so the D′_{i} are decomposition groups): if v_{1},…, v_{n} are independent, then so are, by Corollary 2.5, v′_{1},…, v′_{n}, and hence, by Fact 2.6, F′_{i} must be a henselisation of (F′, v′_{i}); and if char F v_{i} = 0 for all i, then as immediate henselian algebraic extension, F′_{i} is, again, a henselisation of (F′_{i}, v′_{i}).

Now γ′ := res : G_{F′} → Gal(L(A)/L(A)^{A}) = A is a homomorphism with

γ′ | D′_{i} = res : D′_{i} → → Gal(L(A)/(F′_{i} ∩ L(A)) = im β_{i},

so

β_{i}^{−1} ∘ γ′ | D′_{i} = α′ | D′_{i} = res : D′_{i} → → Gal(L/(F_{i} ∩ L)) = α(D_{i}) ≤ B.

Hence, γ′ is a locally exact solution of the lifted locally split embedding problem.□

Proof of Proposition 3.1: The proof of the proposition is now a standard chain construction. We may assume from the start that (F, v_{1} …, v_{n}) etc. satisfies the conditions in the remark following the proposition. By Lemma 3.3, we may also assume that v_{1},…, v_{n} are independent (this is not necessary if char F v_{i} = 0 for all i).

We first find an immediate absolutely defectless extension (F^{1}, v_{1}^{1},…, v_{n}^{1})/ (F, v_{1},…, v_{n}) with ♯F^{1} = ♯F and with decomposition subgroups D_{1}^{1},…, D_{n}^{1} of G_{F^{1}} w.r.t. v_{1}^{1} …, v_{n}^{1} respectively such that G_{F^{1}} =< D_{1}^{1},…, D_{n}^{1} >, such that res : D_{i}^{1} → D_{i} is an isomorphism for each i, and such that each locally split embedding problem for G_{F} w.r.t. D_{1},…, D_{n} has a locally exact solution when lifted to G_{F^{1}} w.r.t. D_{1}^{1},…, D_{n}^{1}.

This is achieved by an ordinal enumeration of these embedding problems (EP_{κ})_{κ<λ} and constructing an (ordinal) chain of absolutely defectless immediate extensions (F_{κ}, v_{κ,1},…, v_{κ,n})/(F, v_{1} …, v_{n}) with ♯F_{κ} = ♯F, with decomposition subgroups D_{κ,i} and isomorphisms res : D_{κ,i} → D_{i} (i = 1,…, n) such that all EP_{μ} for μ < κ have a solution when lifted to G_{F_{κ}}. For successor ordinals this is done by Lemma 3.5, and for limit ordinals by taking unions of the fields constructed ‘before’ (♯F_{κ} never increases since there are only ♯F -many locally split embedding problems for G_{F}). Note that for μ < ν < κ, solutions for EP_{μ} lift from G_{F_{ν}} to G_{F_{κ}}. Then (F^{1}, v_{1}^{1},…, v_{n}^{1}) := (F_{λ}, v_{λ,1} …, v_{λ,n}) has all the required properties.

Now we iterate this process and construct F^{2}, F^{3},… (with valuations, decomposition groups, same cardinality etc.) solving all locally split embedding problems for G_{F^{i}} in G_{F^{i+1}} and let F′ = ⋃_{i=1}^{∞} F^{i} (with v′_{1},… etc.). Then ♯F′ = ♯F, G_{F′} =< D′_{1},…, D′_{n} > and each locally split embedding problem for G_{F′} w.r.t. D′_{1},…, D′_{n} has a locally exact solution. Hence, by Proposition 1.2, G_{F′} = D′_{1} ⋆ ⋯ ⋆ D′_{n}.□

As a consequence, we shall now prove a variant of Proposition 3.1, dropping details about immediacy and absolute defect both from hypothesis and conclusion, but retaining the Galois theoretic data. The reduction of Proposition 3.7 to Proposition 3.1 proceeds via the following

Lemma 3.6 Given a field K, there is an absolutely defectless henselian valued field (L, w) with ♯L = max{♯K, ℵ_{0}} admitting an immediate extension (L̂, ŵ) with ♯L̂ > ♯L such that K ⊆ O_{w}, Lw is the perfect hull of K, Γ_{w} is divisible, and, hence, res : G_{L} → G_{K} is an isomorphism.

Proof: Choose an infinite set X of indeterminates over K with ♯X = max{♯K, ℵ_{0}} and fix some well-ordering ‘<’ on X. Let L = K(X) and let w be the ‘(X, <)- adic’ valuation on L: For any finite subset {x_{1} < k_{2} < … < x_{n}} ⊆ X, the restriction of w to K(x_{1},… x_{n}) is (equivalent to) the composed valuation w_{x_{1}} ⊕ ⋯ ⊕ w_{x_{n}}, where w_{x_{i}} is the x_{i}-adic valuation on the rational function field K(x_{i+1},…, x_{n})(x_{i}) in x_{i} over K(x_{i+1},…, x_{n}). To make this consistent, define

Γ_{w} := ⨁_{x∈X} Z · γ_{x},

where ‘⨁’ is the lexicographic sum w.r.t. the (well-)ordering induced by < on X under the bijection (x ↦ γ_{x})_{x∈X}, define w(x) := γ_{x} for all x ∈ X and define w to be trivial on K. This uniquely determines a valuation w on L with residue field K and value group Γ_{w}.

Now the field of formal Laurent series in X over K

L̂ := K((X)) := {α = ∑_{γ∈Γ_{w}} a_{γ}t^{γ} | a_{γ} ∈ K & supp(α) is well-ordered}

(where supp(α) := {γ ∈ Γ_{w} | a_{γ} ≠ 0}) with the canonical henselian valuation

ŵ(α) := min{γ | a_{γ} ≠ 0} for all α = ∑_{γ∈Γ_{w}} a_{γ}t^{γ} ∈ L̂

is an immediate extension of (L, w). (Henselianity of ŵ is proved, e.g., in [PrC] II.5 Satz 4 and III.2 Satz 17.) Moreover, ♯L̂ > ♯L, since any subset Y ⊆ X is well-ordered and, thus, gives an element

α_{Y} := ∑_{x∈Y} x = ∑_{x∈Y} t^{γ_{x}} ∈ L̂,

where α_{Y} ≠ α_{Y′} for Y ≠ Y′ ⊆ X. So ♯L̂ ≥ ♯{Y ⊆ X} > ♯X = ♯L.

Finally, replace (L̂, ŵ) by the fixed field of the complement of the inertia subgroup of GL̂, and replace (L, w) by its relative algebraic closure in the new (absolutely defectless henselian) field (L̂, ŵ). Then, by Observation 2.3, (L, w) is absolutely defectless and henselian, (L̂, ŵ)/(L, w) is still immediate, now with divisible value group and the perfect hull of K as residue field. Since K ⊆ O_{w}, there is a commutative diagram

G_{L} →^{≅} G_{Lw}
res ↓  ↓≅ res
G_{K} = G_{K}

and so res : G_{L} → G_{K} is an isomorphism as well.□

It may be worth noting that it was only by the special choice of Γ_{w} that ♯K((Γ_{w})) > ♯K + ♯Γ_{w}. If Γ_{w} = R, for example, ♯K((R)) = ♯K + ♯R, since well-ordered subsets of R are countable.

We conclude this section by a variant of Proposition 3.1:

Proposition 3.7 Let (F, v_{1},…, v_{n}) be an n-fold valued field and let D_{1},…, D_{n} be decomposition subgroups of G_{F} w.r.t. v_{1},…, v_{n} respectively. Then there is an extension (F′, v′_{1},…, v′_{n})/(F, v_{1},…, v_{n}) with ♯F′ = max{♯F, ℵ_{0}} such that G_{F′} ≅ D′_{1} ⋆ ⋯ ⋆ D′_{n}, where for each i, D′_{i} is a decomposition subgroup of G_{F′} w.r.t. v′_{i}, and res : D′_{i} → D_{i} is an isomorphism.

Remark 3.8 The extension of n-fold valued fields established in the proposition is, in general, no longer immediate, but (as the proof will show), there is a divisible ordered abelian group Γ such that, for each i,

Γ_{v′_{i}} = Γ ⊕_{lex} p^{−∞}Γ_{v_{i}},

where p = char F (not necessarily char F v_{i}!) and p^{−∞}Γ_{v_{i}} denotes the pdivisible hull of Γ_{v_{i}} (:= Γ_{v_{i}} if p = 0), and F′v′_{i} = F v_{i} or the perfect hull of F v_{i}, again depending on whether or not char F = 0.

Proof: Let (L, w) be the field constructed in the previous lemma from K = F, and let w_{1} = … = w_{n} = w. By Proposition 3.1, there is an immediate extension (L′, w′_{1},…, w′_{n})/(L, w_{1},…, w_{n}) with ♯L′ = ♯L = max{♯F, ℵ_{0}} such that G_{L′} = D_{w′_{1}} ⋆⋯⋆D_{w′_{n}}, where for each i, D_{w′_{i}} is a decomposition subgroup of G_{L′} w.r.t. w′_{i}, and ρ_{i} := res : D_{w′_{i}} → D_{w_{i}} = G_{L} ≅ G_{F} is an isomorphism.

Now consider for each i, the composed valuation w′_{i}⊕v_{i}, i.e. the refinement of w′_{i} by the unique prolongation of (again denoted by) v_{i} from F to the perfect hull L′w′_{i} ≅ Lw_{i} = Lw of F. Let D′_{i} := ρ_{i}^{−1}(D_{i}), let F′_{i} be the fixed field of D′_{i} and let F′ = F′_{1} ∩… ∩ F′_{n}.

Then F′_{i} is a henselisation of (L′, w′_{i} ⊕ v_{i}), hence of the intermediate field F′ w.r.t. the induced prolongation v′_{i} of w′_{i} ⊕ v_{i} from L′ to F′, and res : D′_{i} := G_{F′_{i}} → D_{i} is an isomorphism. Clearly, ♯F′ = max{♯F, ℵ_{0}}. And since the subgroup of a free product of profinite groups G_{1} ⋆ ⋯ ⋆ G_{n} generated by subgroups H_{i} ≤ G_{i} (i = 1,…, n) is the free product of these subgroups: < H_{1},…, H_{n} >= H_{1} ⋆ ⋯ ⋆ H_{n}, we also have G_{F′} = D′_{1} ⋆ ⋯ D′_{n}.□

4 Proof of Theorem 1

1. It clearly suffices to prove Theorem 1 for n = 2. So we are given two fields F_{1}, F_{2} and we want to find a field F with G_{F} ≅ G_{F_{1}} ⋆ G_{F_{2}}, where char F = char F_{1}, provided char F_{1} = char F_{2}.

2. It is well-known that the absolute Galois group of a field K of characteristic p > 0 can be realized as absolute Galois group of a field L of characteristic 0: just make K the residue field of a valuation of mixed characteristic (extend the p-adic valuation on Q canonically to a valuation on the purely transcendental extension Q(X) of Q with residue field F_{p}(X), where X is a transcendence basis of K over F_{p} and adjoin roots of minimal polynomials of all elements of K over F_{p}(X) lifted to Q(X)), pass to the henselisation L′ of L and then to the fixed field of a complement of the inertia subgroup of G_{L′} (use Fact 2.1(a)). Hence we may assume that char F_{1} = char F_{2}.

3. Since for any subgroups H_{1} ≤ G_{1} and H_{2} ≤ G_{2} of profinite groups G_{1}, G_{2}, the subgroup generated by H_{1} and H_{2} (under the canonical embeddings of G_{1}, G_{2}) in G_{1} ⋆ G_{2} is H_{1} ⋆ H_{2}, and since subgroups of absolute Galois groups are absolute Galois groups, it suffices to realize G_{k(X)} ⋆ G_{k(Y)} as absolute Galois group of a field of the characteristic of k, where k = Q or F_{p} is the prime subfield of F_{1} and F_{2}, and where X and Y are transcendence bases of F_{1} and F_{2} over k which we may assume algebraically independent.

4. Now choose valuations v_{1} and v_{2} on F = k(X ∪ Y) with F v_{1} = k(X) and F v_{2} = k(Y) and apply Proposition 3.7 to obtain a field with absolute Galois group D_{1} ⋆ D_{2}, where D_{i} is a decomposition subgroup of G_{F} w.r.t. v_{i} (i = 1, 2). Passing to complements of the inertia subgroups of D_{1}, D_{2} we obtain, once more applying the argument in 3., a field with absolute Galois group G_{k(X)} ⋆ G_{k(Y)}.□

By the same arguments as in Step 3 and 4, it is clear that the proof of Theorem 1 can be reduced to realizing G_{k(X)} ⋆G_{k(X)} as absolute Galois group having the same characteristic as the prime field k, where X is in infinite set of indeterminates over k. But we do not know whether G_{k(X)} ≅ G_{k(X)}⋆G_{k(X)}.

Theorem 1 has an almost trivial generalisation to ‘pro-C Galois groups’: Let C be an almost full family of finite groups, i.e. C is closed under homomorphic images, subgroups and direct products. A pro-C group is then an inverse limit of groups in C and the free pro-C product G_{1} ⋆_{C} ⋯⋆_{C} G_{n} of pro-C groups G_{1},…, G_{n} is a pro-C group G admitting embeddings ϵ_{i} : G_{i} → G such that given any homomorphisms γ_{i} : G_{i} → H into a pro-C group H there is a unique homomorphism γ : G → H with γ_{i} = γ ∘ ϵ_{i} (i = 1,…, n). The pro-C Galois group G_{F} (C) of F is the maximal pro-C quotient of G_{F}, i.e. the Galois group of the compositum of all finite Galois extensions of F with Galois group in C.

Corollary 4.1 Given any fields F_{1},…, F_{n}, there is a field F with G_{F} (C) ≅ G_{F_{1}}(C) ⋆_{C} ⋯ ⋆_{C} G_{F_{n}}(C). Moreover, F can be chosen to have the same characteristic as all F_{i}, provided they have the same characteritic.

Proof: This is, because the maximal pro-C quotient of the free product is the free pro-C product of the maximal pro-C quotients.□

5 Regularly closed fields

5.1 Realizing free products over regularly closed fields

Proposition 5.1 Let (F, v_{1},…, v_{n}) be an n-fold valued field with corresponding henselisations F_{1},…, F_{n}. Then there is an extension (F′, v′_{1},…, v′_{n})/ (F, v_{1},…, v_{n}) of n-fold valued fields such that F′ is regularly closed w.r.t. v′_{1},…, v′_{n}, the v′_{i} are independent and, for each i, there is a henselisation F′_{i} containing F_{i} where res : G_{F′_{i}} → G_{F_{i}} is an isomorphism and G_{F′} ≅ G_{F′_{1}} ⋆ ⋯ ⋆ G_{F′_{n}}.

Proof: In the proof of Proposition 3.7 we already constructed an extension (F′, v′_{1},…, v′_{n})/(F, v_{1},…, v_{n}) of n-fold valued fields and, for each i a henselisation F′_{i} of (F′, v′_{i}) containing F_{i} such that res : G_{F′_{i}} → G_{F_{i}} is an isomorphism and G_{F′} ≅ G_{F′_{1}} ⋆ ⋯ ⋆ G_{F′_{n}}. Moreover, by the way how Proposition 3.1 was used in the proof of Proposition 3.7, v′_{1},…, v′_{n} were independent and, for each i, there was an immediate embedding F′_{i} ↪ F̂_{i} of absolutely defectless henselian valued fields where ♯F̂_{i} > ♯F′_{i}, and, again, res : GF̂_{i} → G_{F′_{i}} was an isomorphism.

It now suffices to find a regularly closed extension (F′′, v′′_{1},…, v′′_{n})/ (F′, v′_{1},…, v′_{n}) with F′′ = F′′_{1}∩…∩F′′_{n} where each (F′′, v′′_{i}) embeds (as valued field) into F̂_{i} and where F′′_{i} is the henselisation of (F′′, v′′_{i}) correpsonding to the relative algebraic closure of the embedding of F′′ in F̂_{i}. For then res : G_{F′′_{i}} → G_{F′_{i}} is an isomorphsim for each i, and so res : G_{F′′} → G_{F′} is an isomorphism as well: surjectivity is clear since G_{F′} is generated by the G_{F′_{i}}. And since G_{F′} ≅ G_{F′_{1}} ⋆ ⋯ ⋆ G_{F′_{n}} there is a (unique) homomorphism res^{−1} : G_{F′} → G_{F′′} induced by the local inverses res_{i}^{−1} : G_{F′_{i}} → G_{F′′_{i}} with res ∘ res^{−1} = id_{G_{F′}}. As G_{F′′} is generated by the G_{F′′_{i}}, res^{−1} is surjective and hence res is injective.

To find (F′′, v′′_{1},…, v′′_{n}) one proceeds exactly as in the proof of [HP], Theorem 3.1. By [HP], Theorem 1.8, it suffices to satisfy the local-global principle for rational points on affine plane curves. So using a standard chain argument (as in the proof of Proposition 3.1), the crucial step is to find, given an absolutely irreducible polynomial f(X, Y) ∈ F′[X, Y] with a simple zero (a_{i}, b_{i}) in each F′_{i}, a regular n-fold valued field extension of (F′, v′_{1},…, v′_{n}) of the cardinality of F′ which embeds in each F̂_{i} (as valued field w.r.t. the corresponding prolongation of v′_{i}) and which has a zero of f.

To achieve this, it obviously suffices to embed the function field F′(x, y) (where (x, y) is a generic point of the curve) into each F̂_{i}. But this is easy since ♯F̂_{i} > ♯F_{i}: the point (a_{i}, b_{i}) is simple, say \frac{df}{dY} (a_{i}, b_{i}) ≠ 0; choose ϵ ∈ F̂_{i} \ F′_{i} with v̂_{i}(ϵ) big enough to guarantee that v̂_{i}(f(a_{i} + ϵ, b_{i})) > 2v̂_{i}(\frac{df}{dY} (a_{i} + ϵ, b_{i}); then a′_{i} := a_{i} + ϵ ∈ F̂_{i} is transcendental over F′ and, since F̂_{i} is henselian, there is some b′_{i} ∈ F̂_{i} with f(a′_{i}, b′_{i}) = 0; the embedding of F′(x, y) over F′ into F̂_{i} is now given by mapping (x, y) ↦ (a′_{i}, b′_{i}).□

Corollary 5.2 Let G be a profinite group which is strongly projective relative to subgroups G_{1},…, G_{n} and assume that each G_{i} is isomorphic to an absolute Galois group. Then there is a regularly closed n-fold valued field (F, v_{1},…, v_{n}) with v_{1},…, v_{n} independent and there is an isomorphism ϕ : G → G_{F} such that for each i, ϕ(G_{i}) is a decomposition subgroup of G_{F} w.r.t. v_{i}.

Proof: We will first construct an n-fold valued field (F, v_{1},…, v_{n}) satisfying all stated properties except being regularly closed. To this end let G_{n+1} be a free profinite group with rk G_{n+1} = rk G. Then G_{n+1} is an absolute Galois group as well and we find, as in the proof of Theorem 1, an (n+1)-fold valued field (K, w_{1},…, w_{n+1}) with G_{K} ≅ G_{1} ⋆ ⋯ ⋆ G_{n+1}, where the free factors G_{i} are decomposition subgroups of G_{K} w.r.t. w_{i}, where the w_{i} are independent and where each (K, w_{i}) allows immediate extensions of higher cardinality. Let π : G_{K} → G be an epimorphism identifying the free factors G_{1},…, G_{n} with the correpsonding subgroups of G and projecting G_{n+1} onto G. As G is strongly projective relative to G_{1},… G_{n}, there is a splitting ϕ : G → G_{K} of π with ϕ(G_{i}) conjugate to the factor G_{i} in G_{K} (for i = 1,…, n). Now let (F, v_{1},…, v_{n}) be the fixed field of ϕ(G) where each v_{i} is induced from the henselisation F_{i} of (K, w_{i}) with G_{F_{i}} = ϕ(G_{i}).

Now we continue as in the previous proof. Given a curve C over F with F_{i}-rational points we find immediate prolongations of v_{1},…, v_{n} to the function field L = F (x, y) of C with henselisations L_{i} containing F_{i} such that res : G_{L_{i}} → G_{F_{i}} is an isomorphism for each i. By strong projectivity, again, the epimorphsim res : G_{L} → G_{F} has a splitting ϕ : G_{F} → G_{L}. Let (F′, v′_{1},…, v′_{n}) be the fixed field of ϕ(G_{F}) with v′_{i} induced from henselisations F′_{i} with G_{F′_{i}} = ϕ(G_{F_{i}}). Then res : G_{F′} → G_{F} and res : G_{F′_{i}} → G_{F_{i}} (i = 1,…, n) are isomorphisms and C has an F′-rational point.

Finally, again, a chain of such extensions leads to the desired regularly closed field.□

5.2 The absolute Galois group of regularly closed fields

5.2.1 Solvability of embedding problems is existential

Let us recall that a (finite) embedding problem for a profinite group G is given by a pair of epimorphisms α : G → → B, β : A → → B, (where A and B are finite groups). A solution resp. a proper solution of the embedding problem is a homomorphism resp. an epimorphism γ : G → A such that α = β ∘ γ.

Observation 5.3 Solvability of a finite embedding problem for the absolute Galois group G_{F} of a field F can be expressed by an existential (first-order) formula in the language of fields with parameters from F.

Let α : G_{F} → → B, β : A → → B be the data of a finite embedding problem for G_{F}. Let E := F ix ker α be the fixed field of the kernel of α and let ψ : Gal(E/F) → B be the unique isomorphism making the diagram

G_{F} = G_{F}
res_{F^{sep}/E} ↓  ↓ α
Gal(E/F) →^{ψ} B

commute. We first prove the following Claim: The above embdding problem has a proper solution iff there is a Galois extension L/F containing E and an isomorphism ϕ : Gal(L/F) → A such that the following diagram commutes:

Gal(L/F) →^{ϕ} A
res_{L/E} ↓  ↓ β
Gal(E/F) →^{ψ} B

To prove the claim, assume first that the embedding problem has a proper solution γ : G_{F} → → A. Then the field L := F ix ker γ is a Galois extension of F containing E (since ker γ ⊆ ker α) and there is a unique isomorphism ϕ : Gal(L/F) → A making the top square of the following diagram commute:

G_{F} = G_{F}
res_{F^{sep}/L} ↓  ↓ γ
Gal(L/F) →^{ϕ} A
res_{L/E} ↓  ↓ β
Gal(E/F) →^{ψ} B

Since α = β∘γ and res_{F^{sep}/E} = res_{L/E} ∘res_{F^{sep}/L}, the outer square commutes as well and, hence, so does the bottom square.

For the converse, assume there is a Galois extension L/F containing E and an isomorphism ϕ : Gal(L/F) → A making the bottom square of the diagram above commute. We define γ := ϕ∘res_{F^{sep}/L}, so that the top square commutes. Then γ : G_{F} → A is an epimorphism and

α = ψ ∘ res_{F^{sep}/E} = ψ ∘ res_{L/E} ∘ res_{F^{sep}/L} = (β ∘ ϕ) ∘ (ϕ^{−1} ∘ γ) = β ∘ γ.

The claim is proved.

Our next step is to express the existence of a Galois extension L/F as described in the claim by an ‘almost existential’ formula in the language of fields {+, ×, 0, 1} (allowing parameters from F). We first express that there is a Galois extension L/F with Gal(L/F) ≅ A. This is equivalent to the existence of an irreducible (monic) polynomial f ∈ F [X] of degree d := ♯A such that the F -algebra F [X]/(f) contains d distinct zeros x_{1},…, x_{d} of f, where for each k, the map x_{1} ↦ x_{k} induces the permutation σ_{k} of {x_{1},…, x_{d}} corresponding to the permutation of A = {a_{1} = 1, a_{2},…, a_{d}} given by left multiplication with a_{k}: then L = F (x_{1}) = ⋯ = F (x_{n}) is a Galois extension with Gal(L/F) ≅ A. The elements of the F -algebra F [X]/(f) can be regarded as d-tuples of elements of F

r_{0} + r_{1}X + … + r_{d−1}X^{d−1}/(f) ↦ (r_{0},…, r_{d−1}),

addition is componentwise and multiplication is expressible via polynomials in the coefficients (only depending on the coefficients of f). Therefore, the existence of a Galois extension L/F with Gal(L/F) ≅ A is equivalent to the formula

∃c, x_{1},…, x_{d}, u ∈ F^{d} : Φ(c, x_{1},…, x_{d}, u), where
Φ(c, x_{1},…, x_{d}, u) = ⋀_{k=1}^{d} f_{c}(x_{k}) = 0 ∧ u · ∏_{k≠l}(x_{k} − x_{l}) = 1

∧ ⋀_{k,l,m:a_{k}·a_{l}=a_{m}} x_{l,0} + x_{l,1}x_{k} + … + x_{l,d−1}x_{k}^{d−1} = x_{m} ∧f_{c} is irreducible.

Here x_{l} = (x_{l,0},…, x_{l,d−1}), and for c = (c_{0},…, c_{d−1}) ∈ F^{d}, ‘f_{c}’ denotes the polynomial f_{c}(X) = X^{d} + c_{d−1}X^{d−1} + … + c_{0}. Of course, addition and multiplication on F^{d} occuring in the formula is induced from the F -algebra F [X]/(f_{c}) and 1 = (1, 0,…, 0) ∈ F^{d}.

To express all properties of L/F required by the claim, let E = F (ζ) and let g ∈ F [X] be the irrducible polynomial of ζ over F, say deg g = e, and let ζ_{1} = ζ, ζ_{2},…, ζ_{e} be the conjugates of ζ over F. Then the existence of a Galois extension L/F containing E and of an isomorphism ϕ : Gal(L/F) → A with β ∘ ϕ = ψ ∘ res_{L/E} is equivalent to the formula

∃c, x_{1},…, x_{d}, u, z_{1} …, z_{e} ∈ F^{d} : Φ(c, x_{1},…, x_{d}, u)∧Ψ(c, x_{1},…, x_{d}, z_{1},…, z_{e}),

where

\begin{matrix}Ψ(c, x_{1},…, x_{d}, z_{1},…, z_{e}) = & ⋀_{j=1}^{e} g(z_{j}) = 0 ∧ ⋀_{j≠j′} z_{j} ≠ z_{j′} \\ ∧ ⋀_{j,j′,j′′:ψ^{−1}(β(a_{j′′}))(ζ_{j′})=ζ_{j}} & z_{j′,0} + z_{j′,1}x_{j′′} + … + z_{j′,d−1}x_{j′′}^{d−1} = z_{j} \end{matrix} .

Again, addition and multiplication on F^{d} occuring in the formula are inherited from F [X]/(f_{c}), and z_{j′} = (z_{j′,0},…, z_{j′,d−1}).

The formula which expresses proper solvability of our embedding problem is existential except for the phrase ‘f_{c} is irreducible’ in the formula Φ. It may now come as a minor surprise that the most naive way of making the formula existential — delete ‘f_{c} is irreducible’ — works, provided F is infinite. Yet for finite fields F, G_{F} ≅ Ẑ is projective, and so every embedding problem has a solution (and any formula true in F is equivalent to that truth). Hence, from now on, F is assumed to be infinite.

So let us first assume that the new formula holds for F, say with c, x_{1},…, x_{d}, u, z_{1},… z_{e} ∈ F^{d} witnessing this. Then A may be considered as group of F - algebra automorphisms of the d-dimensional F -algebra

L′ := F [X]/(f_{c}) = F (x_{1}) = ⋯ = F (x_{d}).

For each irreducible factor f of f_{c} over F, the canonical F -algebra epimorphism

π : L′ → → L := F [X]/(f)

maps the d distinct zeros x_{1},…, x_{d} of f_{c} in L′ to d distinct zeros of f_{c} in the field L:

π(u) · ∏_{k≠l}(π(x_{k}) − π(x_{l})) = π(u · ∏_{k≠l}(x_{k} − x_{l})) = π(1) = 1.

Hence, f_{c} is a separable polynomial and

L = F (π(x_{1})) = ⋯ = F (π(x_{d}))

is a Galois extension of F: the splitting field of f_{c} over F. (So all irreducible factors of f_{c} over F generate the same field extension L/F.) Moreover, E ⊆ L, since g(π(z_{1})) = π(g(z_{1})) = 0. It is also easily checked that

G := {a ∈ A | a(kerπ) ⊆ kerπ} = {a ∈ A | f(π(a(X + (f_{c})))) = 0} ≤ A

and that

\begin{matrix}π^{⋆} : G & → & Gal(L/F) \\ a & ↦ & \begin{aligned}σ_{a} : & L & → & L \\ & X + (f) & ↦ & π(a(X + (f_{c}))) \end{aligned} \end{matrix}

is a well-defined isomorphism: note that A acts simply transitive on the zeros of f_{c} in L′, that G acts simply transitive on the zeros of f_{c} in L′ which become zeros of f in L, and so π^{⋆}(G) acts simply transitive on the zeros of f in L. So ϕ := (π^{⋆})^{−1} embeds Gal(L/F) into A, and the formula Ψ implies that β ∘ ϕ = ψ ∘ res_{L/E}.

For the converse, assume L/F is a Galois extension containing E with an embedding ϕ : Gal(L/F) → A such that β ∘ ϕ = ψ ∘ res_{L/E}. Now use Lemma 2.8 and 2.9 to find a tame unramified Galois extension (M, w)/(K, v) of valued fields with Gal(M/K) ≅ A, Mw ≅ L, Kv ≅ F and a primitive element x ∈ M = K(x) over K with irreducible polynomial f ∈ O_{v}[X] over K such that f′(x) ∈ O_{w}^{×}. Then the polynomial f_{c} := \overline{f} ∈ F [X] is separable and the action of A on the zeros of f in M induces an action of A on the zeros of f_{c} in the F -algebra F [X]/(f_{c}) fulfilling our formula:

If w_{1},…, w_{r} are the distinct prolongations of v to M then all zeros of f are in O_{w_{1}} ∩… ∩ O_{w_{r}}, and A permutes the d distinct zeros of f in each O_{w_{j}}. Thus A acts simply transitive on the corresponding (tuples of) zeros of f in the O_{v}-algebra O_{w_{1}} × ⋯ × O_{w_{r}}, and, via the canonical ring epimorphism

O_{w_{1}} × ⋯ × O_{w_{r}} → → Mw_{1} × ⋯ × Mw_{r} ≅ F [X]/(f_{c}),

on the corresponding zeros x_{1},…, x_{d} of f_{c} in F [X]/(f_{c}). In particular, ∏_{k≠l}(x_{k} − x_{l}) is a unit in F [X]/(f_{c}) since ∏_{a_{k}≠a_{l}∈A}(a_{k}(x) − a_{l}(x)) ∈ O_{w_{1}}^{×} ∩ … ∩ O_{w_{r}}^{×}.□

The idea to consider ‘Galois-algebra extensions’ rather than just field extensions when dealing with embedding problems for absolute Galois groups already occurs in Hasse’s 1948-paper [Hs], section 1.

Proposition 5.4 Let (F, v_{1},…, v_{n}) be an n-fold valued field with henselisations F_{1},…, F_{n} ⊆ F^{sep}. Then ‘locally conjugate solvability’ of a finite locally split embedding problem for G_{F} w.r.t. G_{F_{1}},…, G_{F_{n}} can be expressed by an existential formula in the language of n-fold valued fields with parameters from F.

Proof: Let α : G_{F} → → B, β : A → → B, β_{i} : α(G_{F_{i}}) → A (i = 1,…, n) be the data of a finite locally split embedding problem for G_{F} w.r.t. G_{F_{1}},…, G_{F_{n}}, i.e. A, B are finite groups, α, β are epimorphisms, and the β_{i} are homomorphisms with β ∘ β_{i} = id_{α(G_{F_{i}})}.

As in the proof of Observation 5.3, we let E = F ix ker α be the fixed field of the kernel of α, we let ψ : Gal(E/F) → B be the unique isomorphism making the diagram

G_{F} = G_{F}
res_{F^{sep}/E} ↓  ↓ α
Gal(E/F) →^{ψ} B

commute and first prove the following Claim: The above embedding problem has a proper locally conjugate solution iff there is a Galois extension L/F containing E and an isomorphism ϕ : Gal(L/F) → A such that the diagram

Gal(L/F) →^{ϕ} A
res_{L/E} ↓  ↓ β
Gal(E/F) →^{ψ} B

commutes and ϕ^{−1}(im β_{i}) is a decomposition subgroup of Gal(L/F) w.r.t. v_{i} for each i.

‘⇒’: Since a proper locally conjugate solution γ of a locally split embedding problem is, in particular, a proper solution of the embedding problem given by α, β, we can construct L and ϕ as in the proof of Observation 5.3. So we only have to check that, for each i, ϕ^{−1}(im β_{i}) is a decomposition subgroup of Gal(L/F) w.r.t. v_{i}. But this follows since

ϕ^{−1}(γ(G_{F_{i}})) = res_{F^{sep}/L}(G_{F_{i}}) = Gal(L/L ∩ F_{i})

is a decomposition subgroup of Gal(L/F) w.r.t. v_{i} and is conjugate to ϕ^{−1}(im β_{i}) in Gal(L/F).

‘⇐’: For the converse, the claim in the proof of Observation 5.3 already provides an epimorphism γ : G_{F} → → A with α = β ∘γ. Then, for each i, both ϕ^{−1}(im β_{i}) and ϕ^{−1}(γ(G_{F_{i}})) = Gal(L/L ∩ F_{i}) are decomposition subgroups of Gal(L/F) w.r.t. v_{i}. Hence, they are conjugate in Gal(L/F) and so im β_{i} and γ(G_{F_{i}}) are conjugate in A, i.e. γ is a locally conjugate solution and the claim is proved.

Since any solution γ of an embedding problem is a proper solution of the modified embedding problem where A is replaced by im γ, the claim provides also a criterion for locally conjugate solvability of the given locally split embedding problem: it is the criterion of the claim except that ϕ is only required to be an embedding such that, for each i, there is some a_{i} ∈ A with (im β_{i})^{a_{i}} ⊆ im ϕ and with the property that ϕ^{−1}((im β_{i})^{a_{i}}) is a decomposition subgroup of Gal(L/F) w.r.t. v_{i}.

Our next step, again as in the proof of Observation 5.3, is to express proper locally conjugate solvability by an ‘almost existential’ formula in the language of n-fold valued fields, and we may take the formula

∃c, x_{1},…, x_{d}, u, z_{1},…, z_{e} ∈ F^{d} : Φ(c, x_{1},…, x_{d}, u)∧ψ(c, x_{1} …, x_{d}, z_{1},…, z_{e}) from the proof of Observation 5.3 to express existence of a Galois extension L/F containing E and an isomorphism ϕ : Gal(L/F) → A such that ψ ∘ res_{L/E} = β ∘ ϕ. So we only have to express that, for each i, ϕ^{−1}(im β_{i}) is a decomposition subgroup of Gal(L/F) w.r.t. v_{i}.

To achieve this for one i, let K be the fixed field of ϕ^{−1}(im β_{i}) in L. Then K ∩ E is the fixed field of ψ^{−1}(α(G_{F_{i}})) in E, i.e. K ∩ E = F_{i} ∩ E is a decomposition subfield of E/F. Hence v has exactly [K∩E : F] prolongations to E (Fact 2.7 (ii)), and thus at most r := [L : E] · [K ∩ E : F] prolongations to L. Since β_{i} : α(G_{F_{i}}) → A is injective, [L : K] = ♯im β_{i} = ♯α(G_{F_{i}}) = [E : K ∩ E], and so

\begin{aligned}r & = & [L : E] · [K ∩ E : F] = \frac{[L:K∩E]}{[E:K∩E]} · [K ∩ E : F] \\ & = & \frac{[L:K]·[K:K∩E]}{[E:K∩E]} · [K ∩ E : F] = [K : F]. \end{aligned}

Lemma 2.10 now says that K is a decomposition subfield of L/F (i.e. that ϕ^{−1}(im β_{i}) is a decomposition subgroup of Gal(L/F)) w.r.t. v_{i} iff there is a polynomial h(X) = X^{r} +t_{r−1}X^{r−1} +…+t_{0} ∈ F [X] such that h is irreducible over F, has a root in K and satisfies t_{0},…, t_{r−2}, 1 + t_{r−1} ∈ M_{v_{i}}.

Under the identification L ≅ F [X]/(f_{c}) ≅ F^{d} used in the formula already imported from the proof of Observation 5.3, this is expressed, for each i, by the existential formula

∃y_{i} ∈ F^{d}∃h_{i}, w_{i} ∈ F^{r_{i}}Θ_{i}(c, x_{1},…, x_{d}, y_{i}, h_{i}, w_{i}), where
Θ_{i}(c, x_{1},…, x_{d}, y_{i}, h_{i}, w_{i}) = ⋀_{k:a_{k}∈im β_{i}} a_{k}(y_{i}) = y_{i}
∧ ∏_{k≠l∈S_{i}}(a_{k}(y_{i}) − a_{l}(y_{i})) ∈ F^{×}
∧y_{i}^{r_{i}} + h_{i,r_{i}−1}y_{i}^{r_{i}−1} + … + h_{i,0} = 0
∧h_{i,0},…, h_{i,r_{i}−2}, 1 + h_{i,r_{i}−1} ∈ M_{v_{i}}
∧g_{i}(w_{i,0} + w_{i,1}y_{i} + … + w_{i,r_{i}−1}y_{i}^{r_{i}−1} = 0
∧ ⋁_{p,q∈{1,...,d}^{d}} det((z_{1}^{p_{j}}y_{i}^{q_{j}})_{1≤j≤d}) ≠ 0,

where a_{k}(y_{i}) := y_{i,0} +y_{i,1}x_{k} +…+y_{i,d−1}x_{k}^{d−1}, where S_{i} ⊆ {1,…, d} such that the a_{k} with k ∈ S_{i} form a system of coset representatives of A/im β_{i}, where r_{i} := ♯A/♯im β_{i}, and where g_{i} is the irreducible polynomial of a primitive element of F_{i} ∩ E over F. Note that the first two lines of Θ_{i} express that F (y_{i}) is the fixed field of ϕ^{−1}(im β_{i}) and that hence the polynomial X^{r_{i}} + h_{i,r_{i}−1}X^{r_{i}−1} + … + h_{i,0} is the irreducible polynomial of y_{i} over F. Moreover, formulas of the type t ∈ O_{v}^{×} and t ∈ M_{v} are existential in the language of valued fields:

\begin{aligned}t ∈ O_{v}^{×} & ⟺ & t ∈ O_{v} ∧ ∃t′ ∈ O_{v} : t · t′ = 1 \\ t ∈ M_{v} & ⟺ & t ∈ O_{v} ∧ ∃t′ ∉ O_{v} : t · t′ = 1. \end{aligned}

The existential formula expressing locally conjugate solvability of our locally split embedding problem is now

∃c, x_{1},… x_{d}, u, y_{1},…, y_{n}, z_{1},…, z_{e} ∈ F^{d}∃h_{1}, w_{1} ∈ F^{r_{1}} … ∃h_{n}, w_{n} ∈ F^{r_{n}} :

Φ′(c, x_{1},…, x_{d}, u)∧Ψ(c; x_{1},…, x_{d}, z_{1},…, z_{e})∧⋀_{i=1}^{n} Θ_{i}(c, x_{1},…, x_{d}, y_{i}, h_{i}, w_{i}),

where Φ′ is the formula Φ except that the phrase ‘f_{c} is irreducible’ is deleted. Note that we may assume F to be infinite since otherwise all valuations are trivial and hence any locally split embedding problem trivially solvable.

If the existential formula holds, then we obtain a solution of our locally split embedding problem, as in the proof of Observation 5.3, by considering the F -algebra L′ := F [X]/(f_{c}), an irreducible factor f of f_{c} over F, and the F -algebra epimorphism π : L′ → → L := F [X]/(f) onto the Galois extension

L/F with Gal(L/F) ↪^{ϕ} A ≤ Aut_{F} L′, where β ∘ ϕ = ψ ∘ res_{L/E}. We have

to check that, for each i, im ϕ contains a conjugate of im β_{i} in A, and that the corresponding subgroup of Gal(L/F) is a decomposition subgroup of Gal(L/F).

So let us focus on one i, and observe that the polynomial h_{i} ∈ F [X] of degree r_{i} has r_{i} distinct roots in L′: the elements a_{k}(y_{i}) with k ∈ S_{i} and that the condition in the second line of Θ_{i} is carried over to L by π. Hence, h_{i} has r_{i} distinct zeros in L. It is easily checked that exactly one irreducible factor h of h_{i} over F is of the shape X^{r} + t_{r−1}X^{r−1} + … + t_{0} with t_{0},…, t_{r−2}, 1 + t_{r−1} ∈ M_{v_{i}}. So we may assume (after replacing y_{i} by a suitable a_{k}(y_{i}) — this is fixed by a subgroup conjugate to im β_{i} in A, namely (im β_{i})^{a_{k}}) that h(π(y_{i})) = 0.

Then K := F (π(y_{i})) contains a zero of g_{i} (i.e. a decomposition subfield of E/F w.r.t. v_{i}), and is itself (by Hensel’s lemma applied to h) contained in some henselisation of (F, v_{i}). Hence K ∩ E is a decomposition subfield of E w.r.t. v_{i}. Moreover, the last line of Θ_{i} means that L′ = F (z_{1}, y_{i}), so

L = π(L′) = π(F (z_{1}))π(F (y_{i})) = EK.

Thus, res : Gal(L/K) → Gal(E/E ∩ K) is an isomorphism and K is a decomposition subfield of L/F. Finally, ϕ(Gal(L/K)) ≤ im β_{i}, as Gal(L/K) fixes π(y_{i}), and since

♯Gal(L/K) = ♯Gal(E/E ∩ K) = ♯α(G_{F_{i}}) = ♯im β_{i},

equality holds.

Conversely, assume that L/F is a Galois extension containing E, that ϕ : Gal(L/K) → A is an embedding with β∘ϕ = ψ∘res_{L/E} and that, for each i, there is some a_{i} ∈ A such that (im β_{i})^{a_{i}} ⊆ im ϕ and such that ϕ^{−1}((im β_{i})^{a_{i}}) is a decomposition subgroup of Gal(L/F) w.r.t. v_{i}. Once again quoting from the proof of Observation 5.3, we find a tame unramified Galois extension (M, w)/(K, v) of valued fields with Gal(M/K) ≅ A, Mw ≅ L, Kv ≅ F and a primitive element x ∈ M = K(x) over K with irreducible polynomial f ∈ O_{v}[X] over K such that f′(x) ∈ O_{w}^{×} and such that the action of A on the distinct zeros of the polynomial f_{c} := \overline{f} ∈ F [X] in the F -algebra F [X]/(f_{c}) satisfies the formulas Φ′ and Ψ.

So only Θ_{i} remains to be checked for each i. To this end, choose y ∈ L such that F (y) is the fixed field of (im β_{i})^{a_{i}} in L and the irreducible polynomial of y over F is of the form X^{r} + t_{r−1}X^{r−1} + … + t_{0} with t_{0},…, t_{r−2}, 1 + t_{r−1} ∈ M_{v_{i}} (Lemma 2.10). Let w_{1},…, w_{s} be the distinct prolongations of v to M, choose distinct a_{2},…, a_{s} ∈ M_{v_{i}} \ {0}, and, by Fact 2.1(iii), choose ỹ_{i} ∈ M with

ỹ_{i} ∈ (y + M_{w_{1}}) ∩ (a_{2}y + M_{w_{2}}) ∩… ∩ (a_{s}y + M_{w_{s}}).

Then K(ỹ_{i}) is the fixed field of (im β_{i})^{a_{i}} in M and the irreducible polynomial h̃_{i} ∈ K[X] of ỹ_{i} over K has coefficients in O_{v} and the induced polynomial h_{i} := \overline{h̃_{i}} over F has coefficients h_{i,0},…, h_{i,r_{i}−2}, 1 + h_{i,r_{i}−1} ∈ M_{v_{i}}. Moreover, h_{i} has r_{i} distinct zeros in L (which gives the second line of Θ_{i}). And, finally, L′ = F (z_{1}, y_{i}) since M = K(z̃_{1}, ỹ_{i}), where z̃_{1} ∈ O_{w} is a lifting of z_{1} in M.□

As an application of independent interest let us mention the following Corollary, which might be helpful for the question whether the inverse Galois problem for Q is decidable: note that there is still a chance that the existential theory of Q be decidable.

Corollary 5.5 Let F be a number field and let F_{1},…, F_{n} be henselisations w.r.t. n distinct primes on F. Then locally conjugate solvability of a locally split embedding problem for G_{F} w.r.t. G_{F_{1}},…, G_{F_{n}} is a diophantine property, i.e. equivalent to an existential first-order formula in the language of fields.

Proof: This is just a combination of our Proposition 5.4 with Rumely’s existential definability of valuation rings in number fields ([Ru]).□

Another immediate consequence is the following

Corollary 5.6 Let (F, v_{1},…, v_{n}) and (F′, v′_{1},…, v′_{n}) be n-fold valued fields which are elementarily equivalent in the the language of n-fold valued fields. Then G_{F} is the free product of suitable decomposition subgroups (w.r.t. v_{1},…, v_{n}) iff the same is true for G_{F′}.

Proof: This is immediate from the claim in the proof of the above proposition together with Proposition 1.3.□

5.2.2 Proof of Theorem 2

In order to prove Theorem 2 we will use the following model theoretic characterisation of regularly closed fields:

Fact 5.7 An n-fold valued field (F, v_{1},…, v_{n}) with pairwise independent valuations and with corresponding henselisations F_{1},…, F_{n} is regularly closed w.r.t. v_{1},…, v_{n} iff (F, v_{1},…, v_{n}) is existentially closed (in the language of n-fold valued fields) in any regular extension (F′, v′_{1},… v′_{n}) provided each F_{i} is existentially closed in some (any) henselisation F′_{i} of (F′, v′_{i}) containing F_{i} (in the language of valued fields).

Proof: This is Theorem 1.9 together with Theorem 4.1 of [HP]: note that the general assumption made in [HP] that F should be of characteristic 0 does not enter the proof of those two Theorems.□

Proof of Theorem 2: Let (F, v_{1},…, v_{n}) be a regularly closed n-fold valued field where v_{1},…, v_{n} are pairwise independent, and let F_{1},…, F_{n} be corresponding henselisations. In order to show that G_{F} is relative projective w.r.t G_{F_{1}},…, G_{F_{n}}, we have to show, by Proposition 1.4, that any finite locally split embedding problem for G_{F} w.r.t G_{F_{1}},…, G_{F_{n}} has a locally conjugate solution.

So let α : G_{F} → → B, β : A → → B, β_{i} : α(G_{F_{i}}) → A (i = 1,…, n) be the data of such a finite locally split embedding problem. Using the above fact together with Proposition 5.4, it suffices to find an extension (F′, v′_{1},…, v′_{n})/(F, v_{1},…, v_{n}) and, for each i, a henselisation F′_{i} of (F′, v′_{i}) containing F_{i}, such that F′/F is regular, F_{i} is existentially closed in F′_{i} (in the language of valued fields) and such that the locally split embedding problem (lifted via res : G_{F′} → → G_{F} to G_{F′}) has a locally conjugate solution.

To achieve this, we have to adjust the proof of Lemma 3.5 to our situation. Let, for each i, F_{i}^{⋆} ⪰ F_{i} be an elementary extension (of valued fields) such that ♯F_{i}^{⋆} > ♯F_{i}. Let L be the fixed field of ker α in F^{sep}, and, after the canonical identification, Gal(L/F) = B, α = res : G_{F} → Gal(L/F) = B and L^{α(G_{F_{i}})} = L ∩ F_{i}.

Consider the field L′ := L(A) with the A-action of Lemma 3.4, and let F′ := L(A)^{A} be the fixed field under this action. Then L′/F′ is a Galois extension with Galois group A, F′/F is regular, and, for each i, L^{′ im β_{i}}/L∩F_{i} is purely transcendental, so we may consider L^{′ im β_{i}} as subfield of F_{i}^{⋆} and denote the induced valuation on F′ by v′_{i}. Now let F′_{i} be the relative algebraic closure of F′ in F_{i}^{⋆} and observe that then F_{i} ⊆ F′_{i}, that F_{i} is existentially closed in F′_{i} (because it is in F_{i}^{⋆}) and that L′ ∩ F_{i}^{⋆} = L^{′ im β_{i}}. Therefore, res : G_{F′} → A = Gal(L′/F′) gives rise to a solution of our lifted locally split embedding problem, which, in particular, is a locally conjugate solution.□

References

[En] O.Endler: Valuation theory, Springer (1972).

[Er1] Y.Ershov: Semilocal fields, Soviet Math.Dokl. 15 (1974), 424-428.

[Er2] Y.Ershov: Projectivity of absolute Galois groups of RC_{ζ}^{⋆}-fields, Proc. III Int. Conf.Algebra Krasnoyarsk 93, deGruyter (1996), 63-80.

[Er3] Y.Ershov: Free products of absolute Galois groups, Dokl.Math. 56(3) (1997), 915-917.

[Ge] W.-D.Geyer: Galois groups of intersections of local fields, Isr.J.Math. 30 (1978), 382-396.

[Gr] K.W. Gruenberg: Projective profinite groups, Journal London Math.Soc. 42 (1967) 155-165.

[Ha] D.Haran: On closed subgroups of free products of profinite groups, Proc.London Math.Soc. 55 (1987), 266-298.

[HaJ] D.Haran, M.Jarden: The absolute Galois group of a pseudo p-adically closed field, J.Reine Angew.Math. 383 (1988), 147-206.

[Hs] H.Hasse: Existenz und Mannigfaltigkeit abelscher Algebren mit vorgegebener Galoisgruppe über einem Teilkörper des Grundkörpers I, Math.Nachrichten 1 (1948), 40-61.

[He] B.Heinemann: On finite intersections of ‘henselian valued’ fields, manuscr.math. 52 (1985), 37-61.

[HP] B.Heinemann, A.Prestel: Fields regularly closed with respect to finitely many valuations and orderings, Canad.Math.Soc.Conf.Proc. 4 (1984), 297-336.

[J] M.Jarden: Infinite Galois theory, in: M.Hazewinkel (ed.): Handbook of Algebra, vol. 1 (1996).

[K] F.-V.Kuhlmann: Valuation theory of fields, abelian groups and modules, to appear.

[M] O.V.Mel’nikov: On free products of absolute Galois groups, (1997, Russian), English transl. in Siberian Math.J. 40 (1) (1999), 95-99.

[Po1] F.Pop: Classically projective groups and pseudo classically closed fields, Prerpint Math.Inst.Heidelberg 1990.

[Po2] F.Pop: Embedding problems over large fields, Annals of Math. 144 (1996), 1-34.

[PrC] S.Prieß-Crampe Angeordnete Strukturen: Gruppen, Körper, projektive Ebenen, Springer Erg.d.Math 98 (1983).

[Ri] P.Ribenboim: Théorie des valuations, Montreal (1968).

[Ru] R.Rumely: Arithmetic over the ring of all algebraic integers, J.reine angew.Math. 368 (1986), 127-133.

e-mail: Jochen.Koenigsmann@uni-konstanz.de