Elementary characterization of fields by their absolute Galois group^{∗}

Jochen Koenigsmann^{†}

Abstract

We classify all fields whose elementary theory is encoded in their absolute Galois group and vice versa. It turns out that — apart from very few exotic fields, which probably don’t exist — there are three types of such fields: real closed fields, certain p-adically closed fields and very special generalized power series fields over those p-adically closed fields. The classification is based on a Galois characterization of henselian fields, an analysis of the absolute Galois group of henselian fields of mixed characteristic and an account on first-order definablity of valuations in the language of fields.

The paper is organized as follows:

1 Introduction 1.1 Two basic questions of Galois theory 1.2 Fields elementarily characterized by their absolute Galois group 1.3 The main result

2 Small absolute Galois groups 3 Tools from valuation theory

3.1 Preliminaries from valuation theory 3.1.1 Hilbert theory 3.1.2 Coarsenings and refinements 3.1.3 p-henselian fields and canonical valuations

3.2 Galois characterization of henselian fields 3.3 Absolute Galois groups of henselian fields of mixed characteristic 3.4 t-henselian fields 3.5 Definable valuations

4 Proof of the Main Theorem: positive part 4.1 Fields of type 2 4.2 Fields of type 3 4.3 Fields of type 4

5 Proof of the Main Theorem: negative part 5.1 Fields elementarily determined by their absolute Galois group 5.2 Why not R((Z_{(q)}))? 5.3 The end

1

∗This paper is a condensed version of my Habilitationsschrift [Ko4]. I am very greatful to Alexander Prestel for many helpful discussions on this subject.

†Heisenberg-Stipendiat der Deutschen Forschungsgemeinschaft (KO 1962/1-1)

1 Introduction

1.1 Two basic questions of Galois theory

The absolute Galois group G_{K} of a field K is the Galois group of a separable closure K^{sep} of K over K, considered as profinite group. It is the inverse limit of the Galois groups of all finite Galois extensions of K (in K^{sep}). It may be thought of as ‘universal’ Galois group in the sense that any (finite or infinite profinite) group realizable as Galois group over K is a quotient of G_{K}.

Right from the beginning, Galois groups have never been computed for their own sake, but because they, in turn, encode arithmetic information about the underlying fields. The first and maybe still most prominent example, which any undergraduate learns today, is Galois’ discovery of polynomials over the rationals which cannot be solved by radicals, since the Galois group of the correpsonding splitting field is not solvable (in the group theoretic sense).

So in terms of absolute Galois groups, the two basic questions of Galois theory are:

1. Given a field K, what is the structure of G_{K}?

2. What does the structure of G_{K} tell us about K?

A great deal of research is in progress on either question. For the field Q of rational numbers or the field Q(t) of rational functions over Q, for example, an answer to the first question still seems to be out of reach; even the ‘easier’ so-called inverse Galois problem whether every finite group is realizable as Galois group over Q or over Q(t) has now been open for more than 100 years.

1.2 Fields elementarily characterized by their absolute Galois group

This paper addresses the second question. It investigates fields K for which G_{K} encodes all arithmetic properties of K. To make this precise, we define a field K to be elementarily characterized by its absolute Galois group G_{K} if

for all fields F : G_{F} ≅ G_{K} ⟺ F ≡ K.

Here ‘≡’ denotes elementary equivalence in the language of fields.

Examples of fields elementarily characterized by their absolute Galois group are the field R of real numbers and the field Q_{p} of p-adic numbers: for any field F

F ≡ R ⟺^{T arski}  F real closed  ⟺^{Artin−Schreier} G_{F} ≅ Z/2Z ≅ G_{R}
F ≡ Q_{p} ⟺^{[AK],[Er]} F p-adically closed ⟺^{[Ko2],[Ef]}  G_{F} ≅ G_{Q_{p}}

Counterexamples are the field C of complex numbers (the characteristic of K = C is not encoded in G_{K} = 1) or finite fields (G_{F_{q}} ≅ Ẑ ≅ G_{C((t))}).

1.3 The main result

It has been conjectured that fields elementarily equivalent to R or to a finite abelian extension of Q_{p} for some prime p are the only fields elementarily characterized by their absolute Galois group ([Ja], Conjecture p. 298). Yet, there are some more finite extensions of Q_{p} with this property, and, more surprisingly, there is a third, ‘transcendental’ class of such fields, not containing — like the elementary classes of R or Q_{p} — any algebraic extension of Q. Here is our classification:

Main Theorem A field K is elementarily characterized by G_{K} iff K is elementarily equivalent to one of the following:

1. R

2. a finite extension L of Q_{p} with ([L : L^{ab}], \frac{p}{p−1}♯μ_{L}) = 1, where p is some prime, L^{ab} is the maximal abelian subextension of L/Q_{p}, and μ_{L} is the group of roots of unity of L

3. L((Z_{(q)})), the generalized power series field over L with exponents from Z_{(q)} = Q ∩ Z_{q}, where L is as in 2. and q is some prime (= p or ≠ p)

4. L((Z_{(p)})), where L is a field such that

(i) char L = 0
(ii) tr.deg. L/Q < ∞
(iii) L admits no proper abelian extension
(iv) L has a henselian valuation with residual characteristic p
(v) L is elementarily characterized by G_{L} only among all fields

of characteristic ≠ p

(vi) cd_{p} G_{L} = 1

5. a field L elementarily characterized by G_{L} such that

(i) G_{L} is not pro-solvable
(ii) ∀ fields F : G_{F} ≅ G_{L} ⇒ char F = 0 & tr.deg. F/Q = ∞

Conjecturally, there are no fields of type 4 and 5.

2 Small absolute Galois groups

A profinite group G is called small if for each n ∈ N there are only finitely many subgroups of G of index n. For example, any (as topological group) finitely generated profinite group is small. A small profinite group is up to isomorphism determined by the isomorphism classes of its finite quotients. This property will imply that if a field K is elementarily characterized by G_{K} then G_{K} is small:

Observation 2.1 ([Kl]) For any field K:

G_{K} is small ⟺ ∀ fields F : (F ≡ K ⇒ G_{F} ≅ G_{K}).

Proof: ‘⇒’: Realizability of a given finite group as Galois group over K (i.e. as finite quotient of G_{K}) can easily be expressed by a first-order sentence in the language of fields. So if G_{K} is small and F ≡ K then G_{F} is also small and has the same finite quotients as G_{K}. Hence G_{F} ≅ G_{K}.

‘⇐’: If G_{K} is not small, say with κ ≥ ℵ_{0} many subgroups of index n, any λ-saturated elementary extension F ⪰ K with λ > κ has more than κ distinct extensions of degree n in F^{sep}, so G_{F} ≅ G̸_{K}.□

Calling a field K elementarily determined by G_{K} if

∀ fields F : (G_{F} ≅ G_{K} ⇒ F ≡ K),

one immediately obtains

Corollary 2.2 A field K is elementarily characterized by G_{K} iff G_{K} is small and K is elementarily determined by G_{K}.

We do not know of any field K that is elementarily determined, but not characterized by G_{K}.

3 Tools from valuation theory

As the classification suggests, the proof of the main theorem requires a good deal of valuation theory. Indeed, the chief ingredient for the ‘positive’ part, that all the fields listed are elementarily characterized by their absolute Galois group, is the fact that (with few exceptions) valuations — like orderings — on a field K are encoded in G_{K}. Building on earlier work in this direction ([Ko2], [EN], [EK], [Ko4], ch.3), we have meanwhile come to a rather simple account of this fact (Theorem 3.1).

But also for the ‘negative’ part of the main theorem, that there are no other fields elementarily characterized by their absolute Galois group, the techniques for constructing new fields with the same absolute Galois group as a given field are mainly valuation theoretic. So we first recall some

3.1 Preliminaries from valuation theory

3.1.1 Hilbert theory

For a valuation v on a field K we denote the value group, the valuation ring, its maximal ideal and its residue field by Γ_{v}, O_{v}, M_{v} and Kv = O_{v}/M_{v} respectively. As a reference for general valuation theory we refer to [En],

[Rm] or [Ku]. In this section we only summarize well-known facts. If (K, v) is a valued field and if L/K is a Galois extension then all pro-

longations of v to L are conjugate over K. Let us fix one such prolongation w. Then the decomposition, inertia and ramification subgroups Z, T and V of the Galois group G = Gal(L/K) are defined as

Z = Z(w/v) = {σ ∈ G | σO_{w} = O_{w}}
T = T (w/v) = {σ ∈ G | ∀x ∈ O_{w} : σx − x ∈ M_{w}}
V = V (w/v) = {σ ∈ G | ∀x ∈ O_{w} : σx − x ∈ xM_{w}}

V and T are normal subgroups of Z with V ≤ T such that

(a) Z/T ≅ Aut(Lw/Kv), the isomorphism being canonically induced by the projection O_{w} → Lw,

(b) T/V ≅ Hom(Γ_{w}/Γ_{v}, μ_{Lw}), where μ_{Lw} denotes the group of roots of unity in Lw, which implies

(c) T/V is abelian and V is the unique Sylow-p-subgroup of T, where p = char Kv; in particular, V = {1} for p = 0.

If L is a separable closure of K, one obtains in addition

(d) V and T have complements in Z (by [KPR]), (e) for primes q ≠ p,the Sylow-q-subgroups of T/V are isomorphic to Z_{q}^{d}, where d = dim_{F_{q}}Γ_{v}/qΓ_{v} = cd_{q}T, the q-th cohomological dimension of T,

(f) If V = {1} then Z ≅ T XI G_{Kv}, where the action of G_{Kv} on T is uniquely determined by the cyclotomic character G_{Kv} → Aut(μ_{Kv^{sep}}).

3.1.2 Coarsenings and refinements

Two valuations v and w are called comparable if one valuation ring contains the other. v is a refinement of w (or w a coarsening of v) if O_{v} ⊆ O_{w} or, equivalently, M_{w} ⊆ M_{v}. In this case, Γ_{w} is canonically isomorphic to a quotient of Γ_{v} by a convex subgroup of Γ_{v}. And, conversely, any such quotient gives rise to a coarsening of v.

If v is a refinement of w, then v induces a valuation v/w on Kw, and an exact sequence

1 → Γ_{v/w} → Γ_{v} → Γ_{w} → 1

of ordered abelian groups. In this situation, v is a henselian valuation iff both w and v/w are henselian valuations.

3.1.3 p-henselian fields and canonical valuations

The material in this section can also be found, for example, in [Ko1] and [EK].

For a field K and a prime p we denote by K(p) the maximal Galois-p-extension of K, that is, the compositum of all finite Galois extensions of K of p-power degree. We write G_{K}(p) := Gal(K(p)/K) for the corresponding Galois group.

A valuation v on K is called p-henselian if it extends uniquely to K(p). This is equivalent to the various forms of Hensel’s Lemma when applied to polynomials over K which split into linear factors over K(p). In fact, it is already equivalent to the condition that v extends uniquely to every Galois extension of degree p. Thus, if char Kv ≠ p and if K contains a primitive p-th root of unity, v is p-henselian iff 1 + M_{v} ⊆ K^{p}.

The relations among all p-henselian valuations on K resemble those among all henselian valuations on a field. Calling a field p-closed if K = K(p), the picture is this:

All p-henselian valuations with non-p-closed residue field are comparable and are coarser than any p-henselian valuation with p-closed residue field. If there are any p-henselian valuations with p-closed residue field then there is a coarsest among those; otherwise there is a finest p-henselian valuation on K.

In either case, we call this distinguished p-henselian valuation the canonical p-henselian valuation. It is comparable to any other p-henselian valuation on K, and, if K ≠ K(p) it is non-trivial iff there is some non-trivial phenselian valuation on K.

For henselian fields we use the same terminology: the canonical henselian valuation on a field K is the coarsest henselian valuation with separably closed residue field if there is any; otherwise it is the finest henselian valuation.

Since the canonical henselian resp. p-henselian valuation on K is comparable to any henselian resp. p-henselian valuation on K, its restriction to a (p-)Galois subfield is still henselian resp. p-henselian. The same is true for any coarsening of the canonical (p-)henselian valuation.

Hence the property of admitting a non-trivial (p-)henselian valuation is inherited by Galois-subfields, provided K is not separably closed resp. K ≠ K(p). As a consequence, it is also inherited by subfields of finite index.

3.2 Galois characterization of henselian fields

Theorem 3.1 ([Ko5], Cor. 2.5, [Ko6], Theorem 1) Let K be a field and assume that, for some prime q, G_{K} has a Sylow-q-subgroup H not isomorphic to Z_{q}, Z/2Z or Z_{2} XI Z/2Z.

Then K admits a henselian valuation with non-q-divisible value group and residual characteristic ≠ q iff H has a non-trivial normal abelian subgroup.

Proof: If K admits a henselian valuation v with Γ_{v} ≠ qΓ_{v} and char Kv ≠ q, then the same holds for the fixed field L of H, and so, by Hilbert theory, the inertia subgroup of H is a non-trivial normal abelian subgroup of H.

Conversely, assume that H has a non-trivial normal abelian subgroup. Since H ≅ Z̸_{q}, this implies that cd_{q} H > 1. Thus, char L = char K ≠ q and L contains a primitive q-th root ζ_{q} of unity. Now the Main Theorems of [EN] (for q = 2) and [EK] (for q > 2) imply that L admits a henselian valuation v with char Lv ≠ q and Γ_{v} ≠ qΓ_{v}.

After dividing Γ_{v} by its maximal q-divisible subgroup, we may assume that v is a coarsening of the canonical henselian valuation on L. We will show that w := v | K is henselian.

Case 1: Lv ≠ Lv(q) and Lv is not real closed Assume w is not henselian. Then there is a finite subextension F of L/K with two distinct prolongations w_{1} = v | F and w_{2} such that ζ_{q} ∈ F and F w_{2} ≠ F w_{2}(q): this is possible as [Lv(q) : Lv] = ∞. By approximation, we find an element x ∈ (1 + M_{w_{1}}) ∩ (a + M_{w_{2}}), where a ∈ O_{w_{2}}^{×} with \overline{a} ∉ F w_{2}^{q}; then F ( ^{q} \sqrt{x})/F is a subextension of L/F of degree q — contradiction.

Case 2: Lv = Lv(q) and v is a rank-1 valuation (i.e. Γ_{v} is archimedean) Assume, again, that w is not henselian and choose a finite subextension F of L/K with two distinct prolongations w_{1} = v | F and w_{2} of w such that ζ_{q} ∈ F. With v, w_{1} and w_{2} are also rank-1 valuations, so we have strong approximation. As char Lv ≠ q and Lv = Lv(q), Γ_{v} (and so Γ_{w} and Γ_{w_{2}}) is not q-divisible. Hence, by strong approximation, we choose x ∈ (1 + M_{w_{1}}) ∩ a(1 + M_{w_{2}}) with a ∈ F^{×} such that w_{2}(a) ∉ qΓ_{w_{2}}. Then w_{2}(x) = w_{2}(a) and so F ( ^{q} \sqrt{x})/F is again a subextension of L/F of degree q — contradiction.

Case 3: Lv = Lv(q) and Γ_{v} has a non-trivial convex archimedean subgroup Let v′ be the coarsening of v obtained by dividing Γ_{v} by this non-trivial convex archimedean subgroup. Since v was assumed to be (a coarsening of) the canonical henselian valuation on L we have Lv′ ≠ Lv′(q) and Lv′ is not real closed. Hence, by case 1, w′ := v′ | K is henselian. Since v/v′ is a henselian rank-1 valuation on Lv′ and since Lv′ is the fixed field of a Sylow-q-subgroup of G_{Kw′}, case 2 implies that w/w′ = (v/v′) | K is also henselian. Thus, w is.

Case 4: Lv = Lv(q) and the only convex archimedean subgroup of Γ_{v} is trivial Then, as in the previous case, any proper coarsening of w is henselian by case 1. But the assumption on Γ_{v} (which passes down to Γ_{w}) implies that O_{w} is the intersection of the valuation rings correpsonding to all proper coarsenings of w. Since they are all henselian, w is too.

Case 5: Lv is real closed Then q = 2. Since Γ_{v} has no non-trivial 2-divisible subgroup, the unique prolongation u of v to L(\sqrt{−1}) is the canonical henselian valuation on L(\sqrt{−1}). Hence, by case 3 and 4, u | F (\sqrt{−1}) is henselian, and, in fact, it is the canonical henselian valuation on F (\sqrt{−1}). So u | F = v | F = w is henselian.□

3.3 Absolute Galois groups of henselian fields of mixed characteristic

If (K, v) is a henselian valued field, then, by Hilbert theory,

\begin{aligned}G_{K} & ≅ & V_{v} XI G_{K}/V_{v} \\ G_{K}/V_{v} & ≅ & T_{v}/V_{v} XI G_{Kv} \end{aligned} ,

where V_{v} and T_{v} are ramification and inertia subgroup of G_{K} w.r.t. v. T_{v}/V_{v} is uniquely determined by Γ_{v}, and the action of G_{Kv} on T_{v}/V_{v} is uniquely determined by the cyclotomic character G_{Kv} → Aut(μ_{Kv^{sep}}). So the structure of G_{K}/V_{v} is well understood. In particular, for any prime q ≠ char Kv, one can easily compute the q-th cohomological dimension of G_{K} ([Me], Theorem 3):

cd_{q}G_{K} = cd_{q}G_{Kv} + dim_{F_{q}} Γ_{v}/qΓ_{v}

Unfortunately, little is known, in general, about V_{v} and the action of G_{K}/V_{v} on V_{v}, except, of course, that V_{v} is a Sylow-p-subgroup of T_{v}, where p = char Kv (V_{v} = 1 for char Kv = 0), and that V_{v} is a free pro-p group if char K = p as well. So the most difficult case seems to be when (K, v) is of mixed characteristic (0, p), i.e. char K = 0 and char Kv = p > 0. One doesn’t even know how, in general, to compute cd_{p}G_{K} from valuation theoretic data.

However, we will see in this section that the difficulties can be overcome if one assumes in addition that G_{K} is small. For then there is a dichotomy in the special case that v admits no non-trivial coarsening of residual characteristic 0, i.e. if O_{v}[\frac{1}{p}] = K: either K is p-adically closed (in the wider sense, i.e. Kv is finite, Γ_{v} a Z-group and v(p) finite, or, equivalently K is elementarily equivalent to a finite extension of Q_{p}) — and here G_{K} is known — or G_{K} ≅ G_{F} for a henselian valued field (F, w) with char F = p and G_{F} /V_{w} ≅ G_{K}/V_{v}. From this, one easily handles the general case, where v may admit non-trivial coarsenings of residual characteristic 0, by taking the finest of them, say v_{0}, so that the valuation induced by v on Kv_{0} falls under the special case and, since V_{v_{0}} = 1, we can compute G_{K} ≅ T_{v_{0}} XI G_{Kv_{0}}.

Proposition 3.2 Let (K, v) be a valued field of mixed characteristic (0, p) and assume that O_{v}[\frac{1}{p}] = K. If K^{×}/(K^{×})^{p} is finite then:

a) Kv is perfect.

b) If Γ_{v} ≠ pΓ_{v} then Γ_{v} ≅ Z and Kv is finite.

Proof: The proof copies essentially that of [Po1], Korollar 2.7.

a) 1 is the only p-th root of unity in Kv, so (1 + M_{v}) ∩ (O_{v}^{×})^{p} = (1 + M_{v})^{p}, and one obtains the canonical exact sequences

1 → O_{v}^{×}/(O_{v}^{×})^{p} → K^{×}/(K^{×})^{p} → Γ_{v}/pΓ_{v} → 1

and

1 → (1 + M_{v})/(1 + M_{v})^{p} → O_{v}^{×}/(O_{v}^{×})^{p} → Kv^{×}/(Kv^{×})^{p} → 1.

Therefore, [Kv^{×} : (Kv^{×})^{p}] ≤ [K^{×} : (K^{×})^{p}] and Kv^{×}/(Kv^{×})^{p} is also finite. This is only possible when Kv is perfect: Either Kv is finite and hence perfect. Or Kv is infinite and hence to any given a ∈ Kv one can find x ≠ y with (a − x^{p}) = (a − y^{p})z^{p} for some z ∈ Kv \ {1}, showing that a = (x^{p} − y^{p}z^{p})(1 − z^{p})^{−1} ∈ Kv^{p}.

b) Let v_{p} be the coarsest coarsening of v with residue characteristic p. As O_{v}[\frac{1}{p}] = K, v_{p} is a rank-1-valuation and, by a), Kv_{p} is perfect. In particular, the valuation v/v_{p} induced by v on Kv_{p} has p-divisible value group. So the assumption Γ_{v} ≠ pΓ_{v} implies Γ_{v_{p}} ≠ pΓ_{v_{p}}.

Under this assumption we will show that Γ_{v_{p}} is discrete (i.e. ≅ Z): if 1 + a_{j} ∈ 1 + M_{v_{p}} (j = 1,…, n) are representatives of an F_{p}-base of (1 + M_{v_{p}})/(1 + M_{v_{p}})^{p}, then for a ∈ M_{v_{p}}, the element 1 + a can be written as

1 + a = (1 + b)^{p} ∏_{j=1}^{n}(1 + a_{j})^{r_{j}} with b ∈ M_{v_{p}} and 0 ≤ r_{j} < p.

Therefore

v_{p}(a) ≥ min_{s+s_{1}+···+s_{n}>0} v_{p}(( \begin{matrix}p \\ s \end{matrix} ) b^{s}a_{1}^{s_{1}} ⋯ a_{n}^{s_{n}}).

If now v_{p}(a) < δ := min{v_{p}(p), v_{p}(a_{1}),… v_{p}(a_{n})}, then v_{p}(a) = v_{p}(b^{p}). Hence any γ ∈]0, δ[ is p-divisible, and so, if ]0, δ[≠ ∅, then Γ_{v_{p}} = {0} ∪ ⋃_{n∈Z} n·]0, δ[ would be p-divisible. Thus ]0, δ[= ∅ and so Γ_{v_{p}} = Z · δ is discrete.

Γ_{v_{p}} ≅ Z implies that Kv_{p} is finite (and hence v = v_{p}): (1 + M_{v_{p}})/(1 + M_{v_{p}})^{p} is finite and, as Γ_{v_{p}} ≅ Z, (1 + M_{v_{p}})^{p} ⊆ 1 + M_{v_{p}}^{2} and there are canonical group epimorphisms

(1 + M_{v_{p}})/(1 + M_{v_{p}})^{p} → → (1 + M_{v_{p}})/(1 + M_{v_{p}}^{2}) → → Kv.□

The next theorem, though purely algebraic, is one of those rare examples of ‘applied model theory’, where the only proof known sofar uses model theory in an essential way.

Theorem 3.3 Let (K, v) be a henselian valued field of mixed characteristic (0, p) with O_{v}[\frac{1}{p}] = K. Assume that G_{K} is small. Then the following are equivalent:

(i) Γ_{v} = pΓ_{v}
(ii) cd_{p}G_{K} ≤ 1
(iii) G_{K} ≅ G_{F} for some henselian valued field (F, v′)with char F = p

Moreover, the isomorphism in (iii) induces isomorphisms V_{v} ≅ V_{v′} and T_{v} ≅ T_{v′}.

Proof: (iii) ⇒ (ii) is clear.

(ii) ⇒ (i) follow from the previous proposition: if G_{K} is small then K^{×}/(K^{×})^{p} is finite; hence Γ_{v} ≠ pΓ_{v} implies that K is p-adically closed and, thus, cd_{p}G_{K} = 2.

(i) ⇒ (iii): Assume Γ_{v} = pΓ_{v}. Let (K^{⋆}, v^{⋆}) ⪰ (K, v) be an ℵ_{1}-saturated elementary extension. Denote the coarsest coarsening of v^{⋆} with residue characteristic p by v_{p}^{⋆}, the finest coarsening with residue characteristic 0 by v_{0}^{⋆} and the residue field of v_{0}^{⋆} by E := K^{⋆}v_{0}^{⋆}.

Then v_{p}^{⋆} induces a (henselian) rank-1-valuation w := v_{p}^{⋆}/v_{0}^{⋆} on E with residue field Ew ≅ K^{⋆}v_{p}^{⋆}. We claim that (E, w) has the following properties:

1.) Γ_{w} ≅ R
2.) Ew is perfect
3.) (E, w) is defectless
4.) G_{E} ≅ G_{Ew}

(Recall that a henselian valued field is called defectless if the fundamental equality ‘e · f = n’ holds for any finite extension.)

ad 1.): As Γ_{v} = pΓ_{v}, we also have Γ_{v^{⋆}} = pΓ_{v^{⋆}} and Γ_{w} = pΓ_{w}. Therefore, considering Γ_{w} as subgroup of R, Γ_{w} is dense in R and hence, by saturation, Γ_{w} = R: given γ ∈ R, we find x ∈ K^{⋆} realizing the type

Φ(x) = {rv^{⋆}(p) < v^{⋆}(x) < sv^{⋆}(p) | r, s ∈ Q with rw(p) < γ < sw(p)}.

So w(x̄) = γ, where ‘¯ ’ denotes the place corresponding to v_{0}^{⋆}. ad 2.): This follows from part a) of the previous proposition, since G_{E} is — as quotient of G_{K^{⋆}} ≅ G_{K} — also small.

ad 3.): Let us first show that (E, w) admits no proper immediate algebraic extension. To this end we consider a proper finite extension E(α)/E. If f ∈ E[X] is the irreducible polynomial of α over E then, by Krasner’s lemma, w(f(E)) is bounded, and, as Γ_{w} ≅ R, we find that sup w(f(E)) =: γ ∈ Γ_{w}.

By saturation, there is some x ∈ E with w(f(x)) = γ: Of course, (E, w) is not saturated. But if z ∈ K^{⋆} realizes the type

Ψ(z) = {rv^{⋆}(p) < v^{⋆}(g(z)) < sv^{⋆}(p) | r, s ∈ Q with rv^{⋆}(p) < v^{⋆}(y) < sv^{⋆}(p)}

where we have chosen y ∈ O_{v_{0}^{⋆}} and g ∈ O_{v_{0}^{⋆}}[X] such that w(ȳ) = γ and ḡ = f (‘¯ ’ denotes again the place corresponding to v_{0}^{⋆}), then for x = z̄: w(f(x)) = γ.

It now follows from theorem 1 and 3 of [Ka] that E(α)/E is not immediate (cf. also [Ku], theorem 7.134). To give a direct argument here, let us assume that E(α)/E is immediate. Then

∀β ∈ E(α)∃y ∈ E with w(y − β) > w(β).

So we find some y ∈ E with w(y − (x − α)) > w(x − α). But then, if α = α_{1},…, α_{n} are the conjugates of α over E, henselianity of w implies that w(z − α_{i}) = w(z − α_{j}) for all z ∈ E, i, j ≤ n and so

w(f(x − y)) = w(∏_{i=1}^{n}(x − y − α_{i})) > w(∏_{i=1}^{n}(x − α_{i})) = w(f(x)).

This contradicts the maximality of w(f(x)). So E(α)/E cannot be immediate after all.

Now the same argument can be repeated replacing E by any finite extension E_{1} of E: if [E_{1} : E] = m, the elements of E_{1} are given by m-tuples of elements of E and the type Ψ(z) becomes an m-dimensional type. Thus no finite extension of (E, w) admits proper immediate algebraic extensions, i.e. (E, w) is defectless.

ad 4.): This is just putting 1.), 2.) and 3.) together: any finite extension of (E, w) is inert, so for any finite Galois extension L/E one has that [L : E] = [Lw : Ew], Lw/Ew is Galois, and Gal(Lw/Ew) ≅ Gal(L/E). Hence G_{E} ≅ G_{Ew}.

To finish the proof consider the field F_{1} := Ew((Γ_{v_{0}^{⋆}})) with its power series valuation v_{1} (so Γ_{v_{1}} = Γ_{v_{0}^{⋆}}). Let F be the fixed field of a complement of V_{v_{1}} in G_{F_{1}} and denote the unique prolongation of v_{1} to F by w_{1}. Then, since Γ_{v_{0}^{⋆}} was p-divisible, still Γ_{w_{1}} ≅ Γ_{v_{0}^{⋆}} and since V_{w_{1}} = V_{v_{0}^{⋆}} = 1 this implies T_{w_{1}} ≅ T_{v_{0}^{⋆}}. Hence, by Hilbert theory,

G_{F} ≅ T_{w_{1}} XI G_{Ew} ≅ T_{v_{0}^{⋆}} XI G_{E} ≅ G_{K^{⋆}} ≅ G_{K},

where the last isomorphism is due to the smallness of G_{K} (Observation 2.1). Note that the action of G_{Ew} on T_{w_{1}} is the same as that of G_{E} on T_{v_{0}^{⋆}}, because the isomorphism G_{E} ≅ G_{Ew} respects the relevant cyclotomic characters, i.e. there is a commutative diagram

G_{E}  ≅  G_{Ew}
↓  ↓
Aut(μ′_{E^{sep}}) ≅ Aut(μ_{Ew^{sep}}),

where μ′_{E^{sep}} denotes the group of all n-th roots of unity with n coprime to p: only they are relevant for the action of G_{E} on T_{v_{0}^{⋆}}, because Γ_{v_{0}^{⋆}} is p-divisible.

Now let v′ be the refinement of w_{1} by v_{p}^{⋆}. Then, by construction,

F v′ ≅ (Ew)v_{p}^{⋆} ≅ K^{⋆}v^{⋆} ≡ Kv,

and for each prime q,

dim_{F_{q}} Γ_{v′}/qΓ_{v′} = dim_{F_{q}} Γ_{v^{⋆}}/qΓ_{v^{⋆}} = dim_{F_{q}} Γ_{v}/qΓ_{v},

where the second equality is, again, due to the smallness of G_{K}. Since, as quotient of G_{K}, G_{Kv} is also small, elementary equivalence implies G_{F v′} ≅ G_{Kv}. And equality of the F_{q}-dimensions gives T_{v′}/V_{v′} ≅ T_{v}/V_{v}. Hence, finally, the isomorphism G_{F} ≅ G_{K} induces isomorphisms T_{v′} ≅ T_{v} and V_{v′} ≅ V_{v}.□

3.4 t-henselian fields

t-henselian fields are fields which topologically behave like henselian fields. We recall that a field topology on a field K is a non-discrete Hausdorff topology on K for which the field operations are continuous. A field topology is called V-topology, if the product of ‘large’ elements is ‘large’, i.e. if the condition

∀U ∈ τ ∃V ∈ τ such that ∀x, y ∈ K (x · y ∈ V → x ∈ U or y ∈ U)

is satisfied, where τ denotes a base of open neighbourhoods of 0. By a wellknown theorem of Kowalsky-Dürbaum and Fleischer (cf. [KD] and [F]), V-topologies on fields are exactly the topologies induced by orderings or non-trivial valuations.

In [PZ2], the notion of a t-henselian field is introduced: a V-topological field (K, τ) is t-henselian if the following topological variant of Hensel’s lemma holds:

∀n ∈ N ∃U_{n} ∈ τ∀u_{0},…, u_{n} ∈ U_{n} ∃x ∈ K : x^{n+2}+x^{n+1}+u_{n}x^{n}+…+u_{0} = 0

Examples (though not the only ones) of t-henselian fields are real closed fields or fields with a non-trivial henselian valuation.

If K is not separably closed, then K admits only one t-henselian topology, and this topology is first-order definable in the language of fields: if f ∈ K[X] is irreducible, separable and of degree > 1 and a ∈ K satisfies f′(a) ≠ 0, then for

U_{f,a} := {f(x)^{−1} − f(a)^{−1} | x ∈ K}

the sets c · U_{f,a} (c ∈ K^{×}) form a base of open neighbourhoods of 0 of the t-henselian topology of K. (All this is from [PZ2], §7 and [Pr], p.202/203.)

And, of course, we can express that this topology is a t-henselian Vtopology by (infinitely many) first-order sentences in the langauge of fields. So any non-separably closed field elementarily equivalent to a t-henselian field is t-henselian. On the other hand, a sufficiently saturated elementary extension of a t-henselian field will always have a non-trivial henselian valuation ([PZ2], Theorem 7.2). So one obtains the following

Corollary 3.4 A non-separably closed field is t-henselian iff it is elementarily equivalent (in the language of fields) to some field admitting a non-trivial henselian valuation.

By imposing additional conditions, one can force a t-henselian field to become henselian:

Lemma 3.5 Let K be a t-henselian field which is neither separably closed nor real closed. If G_{K} is pro-solvable or if K is of finite transcendence degree over its prime field then K admits a non-trivial henselian valuation.

Proof: We first assume that G_{K} is pro-solvable. If for some prime p, K contains a primitive p-th root ζ_{p} of unity, or if char K = p, and if K admits a Galois extension of degree p, and if p = 2, K is not Euclidean, then to admit a non-trivial p-henselian valuation is, by [Ko1], Cor. 2.2, an elementary property of K. As a t-henselian field, K is elementarily equivalent to some field with a non-trivial henselian, in particular p-henselian, valuation. Hence, in this special case, K admits a non-trivial p-henselian valuation.

For the general case we pass, given a prime p dividing ♯G_{K(\sqrt{−1})}, from K to a finite extension F of K(\sqrt{−1}) with ζ_{p} ∈ F (if char F ≠ p) and with F ≠ F (p), where F (p) denotes the maximal Galois-p extension of F. Then F is again t-henselian, and so, by the special case, F admits a non-trivial p-henselian valuation (F cannot be Euclidean as \sqrt{−1} ∈ F).

If G_{F} (or, by Artin-Schreier, equivalently, G_{K(\sqrt{−1})}) is a pro-p group then any p-henselian valuation on F is henselian. Therefore, the canonical henselian valuation on F is non-trivial, and so, by [Po2], Prop. 1.3, its restriction to K is again henselian, as F/K is finite.

If G_{K(\sqrt{−1})} is not a pro-p group, we find a second prime q ≠ p and another finite extension of K(\sqrt{−1}) with non-trivial q-henselian valuation. The restrictions of these two valuations on K(\sqrt{−1}) have henselizations which admit separable extensions of p- resp. q-divisible degree (if L is the henselization of a p-henselian valuation on F then res : G_{L}(p) → G_{F} (p) is an epimorphism). Moreover, as henselizations of K(\sqrt{−1}) they are not formally real. Since G_{K}, and so G_{K(\sqrt{−1})}, was assumed to be pro-solvable, we may apply the Lemma of Neukirch-Pop ([Po1], 1.13) to see that those two valuations on K(\sqrt{−1}) are comparable and the coarser one is henselian. Therefore, the canonical henselian valuation on K(\sqrt{−1}) is non-trivial, and so, again, by [Po2], Prop. 1.3, its restriction to K is a (non-trivial) henselian valuation.

Now assume that K is of finite transcendence degree over its prime field. Choose a prime p dividing ♯G_{K(\sqrt{−1})} and find, as above, a finite extension F_{1} of K(\sqrt{−1}) with ζ_{p} ∈ F_{1} or char K = p, with F_{1} ≠ F_{1}(p) and with non-trivial canonical p-henselian valuation v_{1}. Then for any finite Galois extension F_{2}/F_{1} one has \sqrt{−1}, ζ_{p} ∈ F_{2} and F_{2} ≠ F_{2}(p), so that F_{2} has a non-trivial canonical p-henselian valuation v_{2}.

The restriction of v_{2} to F_{1} extends uniquely to F_{2}: any conjugate of v_{2} on F_{2} over F_{1} is again a p-henselian valuation of F_{2} and hence comparable to the canonical p-henselian valuation v_{2} on F_{2}; this is only possible when it coincides with v_{2}.

So v_{2} | F_{1} extends uniquely to F_{2}(p) and so to F_{1}(p) (⊆ F_{2}(p)). Hence v_{2} | F_{1} is a p-henselian valuation on F_{1} and thus comparable to v_{1}. If we denote by w_{2} the coarser of the two valuations on F_{1}, then w_{2} is a non-trivial coarsening of v_{1} extending uniquely to F_{2}.

Since K is countable we find a sequence of Galois extensions F_{2} ⊆ F_{3} ⊆ … of F_{1} exhausting the separable closure ⋃_{i=1}^{∞} F_{i} of F_{1} and for each i we find a non-trivial valuation w_{i} on F_{1} extending uniquely to F_{i} such that w_{i+1} is a coarsening of w_{i}. As the absolute transcendence degree of F_{1} is finite, the value group of any valuation on F_{1} is of finite rank. Hence there is some n ∈ N such that for all m > n one has w_{m} = w_{n}. This means that w_{n} is a non-trivial henselian valuation on F_{1}, hence the canonical henselian valuation on F_{1} is non-trivial and so, by [Po2] Prop. 1.3, its restriction to K is again non-trivial and henselian.□

3.5 Definable valuations

By Theorem 4.1 of [Ko3], every t-henselian field K which is neither separably closed nor real closed admits a valuation which induces the t-henselian topology and which is definable, i.e. the valuation ring is first-order definable in the language of fields, possibly with parameters from K. The reason is that the canonical p-henselian valuation on a field F with F ≠ F (p) and ζ_{p} ∈ F or char F = p is definable unless p = 2 and F is Euclidean ([Ko3], Theorem 3.1 and its proof). Since K is neither separably closed nor real closed, we find a finite extension F/K with these properties (t-henselian ⇒ p-henselian) and we can talk about F with parameters from K.

Lemma 3.6 Let F be an arbitrary field. Then any henselian rank-1-valuation on F with non-divisible value group is ∅-definable, i.e. definable in the language of fields without parameters from F.

Proof: Let w be a henselian valuation on F of rank 1 with non-divisible value group Γ_{w}. Then F is t-henselian but neither real closed nor separably closed. Hence, by [Ko3], theorem 4.1, there is a (not necessarily ∅-)definable valuation v on F inducing (like w) the unique t-henselian topology on F. Since w is a rank-1-valuation, v must be a refinement of w, so Γ_{w} ≅ Γ_{v}/Δ for some convex subgroup Δ of Γ_{v} and w = π ∘ v, where π : Γ_{v} → → Γ_{w} is the canonical projection.

Now let p be the smallest prime with Γ_{w} ≠ pΓ_{w}. Then we claim that for all γ ∈ Γ_{v}:

γ ∈ Δ ⇔ ∃δ∀ϵ(| ϵ |≤| γ |→ δ + ϵ ∉ pΓ_{v})

‘⇒’: Choose δ ∈ Γ_{v} with π(δ) ∉ pΓ_{w}. Then for any ϵ ∈ Δ = kern π one has π(δ + ϵ) = π(δ) ∉ pΓ_{w} and so δ + ϵ ∉ pΓ_{v}. As Δ is convex this holds for any ϵ with | ϵ |≤| γ |, when γ ∈ Δ. ‘⇐’: Assume γ ∈ Γ_{v} \ Δ, i.e. π(γ) ≠ 0 ∈ Γ_{w}. Given δ ∈ Γ_{v} one then finds α_{0} ∈ Γ_{w} with

π(δ)− | π(γ) |≤ p · α_{0} ≤ π(δ)+ | π(γ) | .

This is so because either Γ_{w} is not discrete and hence pΓ_{w} is dense in Γ_{w}, or Γ_{w} ≅ Z and p = 2. Since at least one of the inequalities is strict one can choose α ∈ π^{−1}(α_{0}) with

δ− | γ |≤ p · α ≤ δ+ | γ |

and so for ϵ := p · α − δ one has | ϵ |≤| γ | and δ + ϵ = p · α ∈ pΓ_{v}.

Hence

∀δ∃ϵ(| ϵ |≤| γ | & δ + ϵ ∈ pΓ_{v})

Therefore, O_{w} is ∅-definable in terms of O_{v}: O_{w} = O_{v} ∪ v^{−1}(Δ), so for any x ∈ F:

x ∈ O_{w} ⇔ \{ \begin{matrix}x ∈ O_{v} or \\ ∃y∀z∀t(z, z^{−1} ∈ xO_{v} ∪ x^{−1}O_{v} → yz ∉ t^{p}O_{v}^{×}) \end{matrix}

And this formula, call it ψ(v, x), works uniformly for all valuations v inducing the t-henselian topology on F.

We already know that there is a parameter-free formula ϕ(x_{1},…, x_{n}; x) in the language of fields such that for certain parameters c = (c_{1},…, c_{n}) ∈ F^{n}, the formula ϕ(c_{1},…, c_{n}; x) defines the valuation ring of the restriction v_{c} of the canonical p-henselian valuation on the extension of F obtained by adjoining a zero of the irreducible polynomial with coefficients c_{1},…, c_{n}. Call this (elementary) property χ(c). Then O_{w} is ∅-definable by the formula

x ∈ O_{w} ⇔ ∀c ∈ F^{n}(χ(c) → ψ(v_{c}, x)).□

In the proof of the lemma we have in fact only used that v induces a t-henselian topology on F. But this makes the lemma not really stronger because any rank-1-valuation with t-henselian topology is already henselian: if v had distinct prolongations to some finite extension L of F, they would induce different topologies on L (rank 1!), so L would not be t-henselian; but like henselianity, t-henselianity goes up finite extensions (which can be seen by passing to an elementarily equivalent henselian field).

We conclude this section with another definability result:

Lemma 3.7 Let F be a field which is not formally real and not separably closed, which is of finite transcendence degree over its prime field and which has small absolute Galois group G_{F}. Then the canonical henselian valuation on F is ∅-definable in the language of fields.

Proof: For each i ∈ N we define F_{i} to be the compositum of all separable extensions of F (in a fixed separable closure F^{sep}) of degree ≤ n. Since G_{F} is small the F_{i} are finite Galois extensions of F and one can talk about them in the language of fields without parameters from F. Of course, F_{i} ⊆ F_{j} for i ≤ j and F^{sep} = ⋃_{i=1}^{∞} F_{i}.

Now assume the canonical henselian valuation v on F is non-trivial (otherwise definability is also trivial). As F is not formally real, the canonical henselian valution on each F_{i} is the unique prolongation of v to F_{i} (again denoted by v) and for any prime p | ♯G_{F} one has F_{i} ≠ F_{i}(p) for all i ≥ some i_{p} ∈ N. For i ≥ i_{p} the canonical p-henselian valuation v_{i} on F_{i} is an ∅-definable refinement of v, and for j ≥ i ≥ i_{p}, v_{j} | F_{i} is a coarsening of v_{i}.

Since F is of finite transcendence degree over its prime field, any valuation on F has only finitely many coarsenings. So for j ≥ some j_{p}, v_{j} | F is henselian as in the proof of Lemma 3.5. If F v is not separably closed then v is the finest henselian valuation on F and so v_{j} = v. And if F v is separably closed then F_{j}v_{j} is also separably closed and v_{j} is the coarsest (p-)henselian valuation on F_{j} with F_{j}v_{j} = F_{j}v_{j}(p). Hence, again v_{j} = v.□

4 Proof of the Main Theorem: positive part

As mentioned in the introduction, fields of type 1 (= real closed fields) are well-known to be elementarily characterized by their absolute Galois group. And, trivially, so are fields of type 5.

So we only have to check that fields K of type 2, 3 and 4 are elementarily characterized by G_{K}. And, in fact, it suffices to show that they are elementarily determined by G_{K}, since G_{K} is small for all of them: If L/Q_{p} is finite, then G_{L} is of rank [L : Q_{p}] + 3 and G_{L((Z_{(q)}))} = Z_{q} XI G_{L} is of rank [L : Q_{p}] + 4; so for K of type 2 or 3, G_{K} is even finitely generated. And for fields K ≡ L((Z_{(p)})) of type 4, G_{L} is small since L is elementarily characterized by G_{L} among all fields of characteristic ≠ p; hence, G_{K} ≅ Z_{p} XI G_{L} is small as well.

4.1 Fields of type 2

Let K be a field of type 2 and let F be a field with G_{F} ≅ G_{K}. Then, by [Ko2], Theorem 4.1 or, for p ≠ 2, also by Pop’s appendix in [Ko2] or by [Ef], K is p-adically closed (in the wider sense of being elementarily equivalent to a finite extension of Q_{p}).

That fields of type 2 are exactly the p-adically closed fields elementarily determined by their absolute Galois group, follows now from straightforward considerations in local class field theory using the observation that two finite extensions K and L of Q_{p} are elementarily equivalent iff they are isomorphic over Q_{p} (finite extensions of Q_{p} are obtained by adjoining zeros of polynomials over Q!) and using the result of [JW], [Rit], [JR], [Di] and [Je] that

G_{K} ≅ G_{L} ⟺ [K : Q_{p}] = [L : Q_{p}] and K^{ab} = L^{ab},

where in case p = 2 the direction ⇐ has only been proved under the additional assumption that K(\sqrt{−1})/K is unramified. We omit the details here (cf. [Ko4], ch.6).

Rather, for the convenience of the reader, we outline a proof of the Galois characterization of p-adically closed fields: So now let K/Q_{p} be any finite extension and let F be a field with G_{F} ≅ G_{K}. We will show that F is p-adically closed.

For any prime q, cd_{q}G_{F} = cd_{q}G_{K} = 2, so char F = 0. If q ≠ p, then the Sylow-q-subgroups of G_{F} are of the shape Z_{q} XI Z_{q}. In particular, they have a non-trivial abelian normal subgroup, and so, by Theorem 3.1, F has a henselian valuation v with Γ_{v} ≠ qΓ_{v} and char F v ≠ q. We may assume v to be a coarsening of the canonical henselian valuation w on F. Hence Γ_{w} ≠ qΓ_{w}.

Since G_{F} has no non-trivial abelian normal subgroup, the ramification subgroup V_{w} of G_{F} w.r.t. w is a non-trivial normal subgroup of G_{F}. It is a pro-l group with l = char F w. But G_{F} has no normal pro-l subgroups except for l = p. So char F w = p.

Now let w_{0} be the finest coarsening of v with char F w_{0} = 0. Then Γ_{w_{0}} is divisible (G_{F} still has no abelian normal subgroups) and G_{F} ≅ G_{F w_{0}}. The valuation w/w_{0} induced by w on F w_{0} is henselian, of mixed characteristic (0, p) and O_{w/w_{0}}[\frac{1}{p}] = F w_{0}. As cd_{p}G_{F w_{0}} = cd_{p}G_{K} = 2, Theorem 3.3 implies that Γ_{w/w_{0}} ≠ pΓ_{w/w_{0}}, and so, by Proposition 3.2, F w_{0} is p-adically closed. Hence, so is F.

4.2 Fields of type 3

Let K = L((Z_{(q)})) be a field of type 3 (with L of type 2), and let F be a field with G_{F} ≅ G_{K}. Then

G_{F} ≅ G_{K} ≅ T_{v} XI G_{L} ≅ Z_{q} XI G_{L},

where v is the power series valuation on K. Let T be the isomorphic copy of T_{v} in G_{F}. Then T is a non-trivial normal abelian subgroup of a Sylowq-subgroup H of G_{F}. Since H ≅ Z̸_{q} or Z/2Z or Z_{2} XI Z/2Z, Theorem 3.1 implies that F has a henselian valuation w with Γ_{w} ≠ qΓ_{w} and char F w ≠ q.

In fact, we can assume that char F w = 0. Otherwise w would only have coarsenings of residual characteristic 0 with divisible value group, because every abelian normal subgroup of G_{F} is contained in T (G_{L} has no nontrivial abelian normal subgroups). So if char F w = l > 0, we might as well assume O_{w}[\frac{1}{l}] = F and thus, from Theorem 3.3 and Proposition 3.2, deduce that F is l-adically closed (cd_{l}G_{F} > 1). This, however, is impossible as cd_{q}G_{F} = cd_{q}G_{K} = 3.

Thus T_{w} = T: as non-trivial abelian normal subgroup of G_{F}, T_{w} is a subgroup of T (as above) and is the inertia subgroup of T w.r.t. the unique prolongation of w to the fixed field of T. Hence T_{w} has a complement in T ≅ Z_{q}; this has to be trivial since T_{w} ≠ 1, so [T : T_{w}] < ∞, but Z_{q} is torsion-free.

So Γ_{w} is divisible by any prime except q, and [Γ_{w} : qΓ_{w}] = q. Dividing Γ_{w} by its maximal convex subgroup, we may, therefore, assume that Γ_{w} is regular and, by [R-Z], Γ_{w} ≡ Z_{(q)}. Moreover,

G_{F w} ≅ G_{F} /T_{w} = G_{F} /T ≅ G_{K}/T_{v} ≅ G_{L},

so since L was elementarily characterized by G_{L}, F w ≡ L. Now, by the Ax-Kochen-Ershov-principle, (F, w) ≡ (K, v), so, in particular, F ≡ K.

4.3 Fields of type 4

Let K = L((Z_{(p)})) be a field of type 4, so L satisfies the conditions (i)-(vi) listed in the Main Theorem under 4. Then L has a henselian valuation v of mixed characteristic (0, p) and L is not separably closed (cd_{p}G_{L} = 1). Since L admits no proper abelian extensions, Γ_{v} is divisible and G_{L} is not prosolvable, so Lv cannot be separbly closed or formally real. We, thus, may assume that v is the canonical henselian valuation, and, since tr.deg.L/Q < ∞ and G_{L} is small, that v is definable (Proposition 3.5 and Lemma 3.7).

Now let F be a field with G_{F} ≅ G_{K}. Then

G_{F} ≅ G_{K} ≅ T_{v′} × G_{L} ≅ Z_{p} × G_{L},

where v′ is the power series valuation on K (note that L contains all roots of unity, so G_{L} acts trivially on T_{v′}).

Since cd_{p}G_{L} = 1, cd_{p}G_{F} = cd_{p}G_{K} = 2, and so char F ≠ p. Hence the fixed field L′ of the isomorphic copy of G_{L} in G_{F} is a field of characteristic ≠ p with G_{L′} ≅ G_{L}. As L is elementarily characterized by G_{L} among all fields of characteristic ≠ p, L′ ≡ L. So the canonical henselian valuation w on L′ is also definable and of mixed characteristic (0, p).

Since G_{L′} is a normal subgroup of G_{F}, w_{F} := w | F is henselian and it is the canonical henselian valuation of F. On the other hand, if T is the isomorphic copy of T_{v′} in G_{F}, it is a non-trivial normal abelian subgroup of a Sylow-p-subgroup H of G_{F}, and H ≅ Z̸_{p} or Z/2Z or Z_{2} XI Z/2Z, as cd_{p}G_{L} = 1. Hence, Theorem 3.1 gives a henselian valuation w′ on F with Γ_{w′} ≠ pΓ_{w′} and char F w′ ≠ p. As we may assume w′ to be a coarsening of w_{F}, char F w′ = 0 and Γ_{w_{F}} ≠ pΓ_{w_{F}}. Yet, Γ_{w} was divisible and Gal(L′/F) ≅ Z_{q}, so Γ_{w_{F}} is divisible by any prime except p, and [Γ_{w_{F}} : pΓ_{w_{F}}] = p.

The same holds then for Γ_{w′} and we can proceed as in the proof for fields of type 3, showing that T = T_{w′} and that we may assume Γ_{w′} to be regular, so Γ_{w′} ≡ Γ_{v′} = Z_{(p)}, and G_{F w′} ≅ G_{L} implying F w ≡ L. Finally, by the Ax-Kochen-Ershov-principle, (F, w′) ≡ (K, v′) and, in particular, F ≡ K.

5 Proof of the Main Theorem: negative part

5.1 Fields elementarily determined by their absolute Galois group

We first collect some simple properties of fields elementarily determined by their absolute Galois group. The crucial observation that they are thenselian is due to Prestel ([Pr]), who was the first to establish some of their general properties.

Proposition 5.1 Let K be elementarily determined by G_{K}. Then

(a) char K = 0,

(b) K is t-henselian,

(c) G_{K} is not projective.

Proof: Like any other field, K can be made the residue field of some nontrivially valued field (L, v) of characteristic zero. Passing to a henselization L^{h} of (L, v) and then to the fixed field F of a complement of the inertia subgroup of G_{L^{h}} in L^{sep}, we obtain a non-trivially henselian valued field F with char F = 0 and G_{F} ≅ G_{K}. This implies (a), and since K cannot be separably closed, also (b) (Corollary 3.4). And, since any projective profinite group can (as subgroup of a free profinite group) be realized as absolute Galois group of fields of arbitrary characteristic, (c) follows as well.□

Within the class of fields of characeristic 0, algebraically closed fields are, of course, also elementarily determined by G_{K}. However, the following lemma shows that in most other cases it makes no difference whether K is elementarily determined by G_{K} among all fields or only among fields of characteristic 0.

Lemma 5.2 Let K be a field allowing non-trivial abelian extensions. Then K is elementarily determined by G_{K} iff char K = 0 and K is elementarily determined by G_{K} among the fields of characteristic 0.

Proof: If K is elementarily determined by G_{K} then, by Proposition 5.1, char K = 0, and all fields F with G_{F} ≅ G_{K}, especially those of characterisitic 0, are elementarily equivalent to K.

Conversely, if char K = 0 and all fields F of characteristic 0 with G_{F} ≅ G_{K} are elementarily equivalent to K, then there is no field L with char L = p > 0 and G_{L} ≅ G_{K}. Otherwise one can assume L to be perfect and find a henselian valued field (F, v) of mixed characteristic (0, p) with residue field L and value group Z, the value of p being 1, and then construct two elementarily non-equivalent algebraic extensions of F with absolute Galois group G_{L}:

By assumption, the field K, and hence L, admits Galois extensions of some prime degree q. If L admits a Galois extension L′/L of degree q = p, we replace F by F (ζ_{p}) (the residue field is still L) and choose maximal purely wild extensions F_{1} resp. F′_{1} of F (ζ_{p}) containing ^{p} \sqrt{p} resp. ^{p} \sqrt{pα}, where a p-th root of α ∈ F (ζ_{p}) generates the unique Galois extension of F (ζ_{p}) of degree p with residue field L′ (note that the extensions F (ζ_{p}, ^{p} \sqrt{p})/F (ζ_{p}) and F (ζ_{p}, ^{p} \sqrt{pα})/F (ζ_{p}) are purely wild). Since by Prop. 4.3 from [KPR], the maximal purely wild algebraic extensions are just the complements of the (absolute) ramification fields, F_{1} and F′_{1} are complements of the ramification subfield of F̃/F (ζ_{p}) (F̃ denotes the algebraic closure of F). Passing now to complements F_{2} and F′_{2} of the inertia subfields of F̃/F_{1} and F̃/F′_{1}, we have G_{F_{2}} ≅ G_{F′_{2}} ≅ G_{L}, but ^{p} \sqrt{p} ∈ F_{2} \ F′_{2} ( ^{p} \sqrt{p} ∈ F′_{2} would imply ^{p} \sqrt{α} ∈ F′_{2}, so the residue field of F′_{2} would have to contain L′, though we know it to be still L). Hence F_{2} ≢ F′_{2}.

If L admits Galois extensions of prime degree q ≠ p, we choose a minimal such prime q and observe that then L and so, by henselianity, F contains ζ_{q}, so for some α ∈ F \ F^{q}, the Galois extension F ( ^{q} \sqrt{α})/F induces a Galois extension of L of degree q. Choose again some complement F_{1} of the ramification subfield of F̃/F, and then two complements F_{2} and F′_{2} of the inertia subfield of F̃/F_{1}, with F_{2} containing ^{q} \sqrt{p} and F′_{2} containing ^{q} \sqrt{pα}. As above, G_{F_{2}} ≅ G_{F′_{2}} ≅ G_{L}, but p ∈ F_{2}^{q} \ (F′_{2})^{q}, so F_{2} ≢ F′_{2}.

This proves that any field with absolute Galois group G_{K} has characteristic 0, so K is indeed elementarily determined by G_{K}.□

We remark that the construction of the two elementarily inequivalent fields in the proof can be carried out in such a way that the two fields have the same p^{n}-th roots of unity.

No answer to the following question seems to be known:

Question 5.3 Is every field K which admits no proper abelian extensions cohomologically trivial (i.e. G_{K} projective)?

A positive answer to this question, at least for fields of finite transcendence degree over Q, would obviously rule out fields of type 4 in our Main Theorem.

Proposition 5.4 Let K be elementarily determined by G_{K} and let v be a henselian valuation on K with char Kv = 0. Then either Γ_{v} is divisible or there is a prime p such that

[Γ_{v} : qΓ_{v}] = \{ \begin{matrix}1 & for all primes q ≠ p \\ p & for q = p \end{matrix}

and Kv is elementarily determined by G_{Kv} and μ_{p^{∞}}(Kv) among fields of characteristic ≠ p.

Proof: We first show that Γ_{v} = qΓ_{v} for all primes q except for at most one prime, say p, where [Γ_{v} : pΓ_{v}] = p.

Let us assume the contrary. Then for some primes p, q there are elements γ, δ ∈ Γ_{v} with γ ∉ pΓ_{v}, δ ∉ qΓ_{v} and, if p = q, γ and δ are F_{p}-independent mod pΓ_{v}. Define

Δ_{1} := Γ_{v}(\{\frac{1}{p^{n}} γ | n ∈ N\})
Δ_{2} := Γ_{v}(\{\frac{1}{p^{n}} γ | n ∈ N\} ∪ \{\frac{1}{q^{n}} δ | n ∈ N\})
Γ_{1} := Z_{(p)} ⊕_{lex} Δ_{1}
Γ_{2} := (Z_{(p)} ⊕ αZ_{(q)}) ⊕_{lex} Δ_{2},

where α ∈ R \ Q, so Z_{(p)} ⊕ αZ_{(q)} ≤ R.

We set F_{i} := Kv((Γ_{i})) (for i = 1, 2) and denote the inertia subgroup of G_{K} w.r.t. v and of G_{F_{i}} w.r.t. the power series valuation on F_{i} by T_{K} and T_{F_{i}} respectively. Then T_{K} ≅ T_{F_{i}}, since for all primes l one has

dim_{F_{l}} Γ_{v}/lΓ_{v} = dim_{F_{l}} Γ_{i}/lΓ_{i} (i = 1, 2).

Moreover, the common residue field Kv has characteristic 0, so it follows that

G_{F_{i}} ≅ T_{F_{i}} XI G_{Kv} ≅ T_{K} XI G_{Kv} ≅ G_{K}.

As K is elementarily determined by G_{K}, F_{1} ≡ F_{2} ≡ K.

Now denote the rank-1-coarsenings of the power series valuation of F_{i} by w_{i}. So Γ_{w_{1}} = Z_{(p)} and Γ_{w_{2}} = Z_{(p)} ⊕ αZ_{(q)}. By Lemma 3.6, w_{1} and w_{2} are ∅-definable. Let ϕ be a parameter-free formula defining w_{1} in F_{1}. Then

F_{2} ≡ F_{1} ⊧ ϕ defines a valuation w

inducing the t-henselian topology

and such that [Γ_{w} : lΓ_{w}] = \{ \begin{matrix}1 & for l = q \\ p & for l = p \end{matrix}

(In the terminology of section 3.4, ‘w induces the t-henselian topology’ is expressed by saying ‘for all irreducible polynomials f of degree p and for all a with f′(a) ≠ 0:

∀x∃y : x · O_{w} ⊆ y · U_{f,a} and ∀x∃y : x · U_{f,a} ⊆ y · O_{w}.)

But F_{2} admits no such valuation w: it would have to induce the same topology as the rank-1-valuation w_{2}, but then [Γ_{w_{2}} : pΓ_{w_{2}}] ≤ p and if p ≠ q, then Γ_{w_{2}} = qΓ_{w_{2}} which is not true for Γ_{w_{2}} = Z_{(p)} ⊕ αZ_{(q)}.

This proves the statement about the value group Γ_{v}.

If Γ_{v} is divisible then G_{Kv} ≅ G_{K} and so also Kv is elementarily determined by G_{Kv}. If Γ_{v} ≠ pΓ_{v}, the action of G_{Kv} on T_{v} depends only on the p-power roots of unity in Kv. So any henselian valued field (F, w) with V_{w} = 1, char F w ≠ p, with Γ_{w} ≡ Γ_{v}, with G_{F w} ≅ G_{Kv} and with μ_{p^{∞}}(F w) = μ_{p^{∞}}(Kv) has the same absolute Galois group as (K, v). Passing to the definable coarsenings of w and v with regular non-divisible value group doesn’t change any of these data. So as F ≡ K, also F w ≡ Kv.□

Corollary 5.5 Let K be a field elementarily determined by G_{K} and assume that either G_{K} is pro-solvable or K is of finite transcendence degree over Q. Then K is real closed, or K admits a non-trivial henselian valuation and the canonical henselian valuation on K is of mixed characteristic.

Proof: By Prop. 5.1, char K = 0, K is not separably closed and K is thenselian. Assuming that K is not real closed, Lemma 3.5 implies that K admits a non-trivial henselian valuation. We denote the canonical henselian valuation on K by v. We have to show that char Kv ≠ 0.

Suppose the contrary is true, i.e. char Kv = 0. Then Γ_{v} is not divisible, since otherwise G_{Kv} ≅ G_{K}, so Kv is not separably closed and hence, v is the finest henselian valuation on K. However, repeating the above argument for Kv in place of K shows that Kv admits a non-trivial henselian valuation thus inducing a proper refinement of v on K. Contradiction.

So Γ_{v} ≠ pΓ_{v} for some prime p. Now the previous proposition shows that [Γ_{v} : pΓ_{v}] = p and that Γ_{v} is divisible by any prime other than p. Hence the inertia subgroup of G_{K} w.r.t. v is isomorphic to Z_{p}, and so G_{K} ≅ Z_{p} XI G_{Kv}. Moreover, Kv is elementarily determined by G_{Kv} and μ_{p^{∞}}(Kv) among all fields of characteristic ≠ p.

As G_{K} is not projective, Kv cannot be separably closed. Hence v is the finest henselian valuation on K. The henselian field F = Kv((Q)) satisfies G_{F} ≅ G_{Kv}, μ_{p^{∞}}(F) = μ_{p^{∞}}(Kv) and char F = 0 ≠ p, so F ≡ Kv and this implies that Kv must be t-henselian (as in Prop.5.1). Since the property that K has a pro-solvable absolute Galois group or that K is of finite transcendence degree over Q is inherited by Kv, Leamma 3.5 implies that either Kv admits a non-trivial henselian valuation (which is impossible as this would, again, induce a proper henselian refinement of v on K), or that Kv is real closed. But then G_{K} ≅ G_{L}, where L = R((Z_{(p)})), and it will be proved in Prop. 5.6 (of course, without using this corollary), that L is not elementarily determined by G_{L}. Therefore, char Kv ≠ 0.□

5.2 Why not R((Z_{(q)}))?

Proposition 5.6 Let q be a prime and let K := R((Z_{(q)})). Then there is a subfield F of Q̃ with G_{F} ≅ G_{K}.

In particular, K is not elementarily determined by G_{K}.

Proof: If q = 2 then any subfield F of Q̃ which is algebraically maximal w.r.t. two orderings has G_{F} ≅ G_{K} ≅ Z_{2} XI Z/2Z ([BEK]).

If q ≠ 2 let F be a maximal ordered subfield of Q̃ such that 2 ∉ F^{q}. Then F^{2} is the unique ordering of F and the only proper minimal algebraic extensions of F are F (i) (where i^{2} = −1) and the q distinct conjugates of F ( ^{q} \sqrt{2}).

Choose some real closure R of F and σ ∈ G_{F} such that σ( ^{q} \sqrt{2}) = ζ_{q} ^{q} \sqrt{2}.

We claim that F = R ∩ σR. Clearly F = F ∩ σF ⊆ R ∩ σR. For the inclusion R ∩ σR ⊆ F it is enough to check that i ∉ R ∩ σR and that no conjugate of ^{q} \sqrt{2} is in R∩σR, as any proper algebraic extension of F contains one of these elements. i ∉ R, so i ∉ R ∩ σR. And because R contains only one q-th root of 2, R ∩ σR contains none, since σ fixes none of them.

But now, by [PZ1], Korollar 3.3 and its proof, F is hereditarily Euclidean, and G_{F} ≅ H XI Z/2Z, where H is torsion free pro-cyclic and odd.

By the maximality of F, H ≅ Z_{q}: the unique Sylow-q-subgroup of H is a characteristic subgroup of H, hence a normal subgroup of G_{F}. Denote its fixed field by L. By the theorem of Schur-Zassenhaus, Gal(F (i)/F) has a complement in Gal(L/F); for the fixed field L′ of this complement, one has 2 |̸ [L′ : F] and q |̸ [L′ : F]; therefore L′ = F, L = F (i) and so H ≅ Z_{q}.

Thus G_{F} ≅ Z_{q} XI Z/2Z with non-trivial action (i.e. by inversion), which implies G_{F} ≅ G_{K}.

The ‘in particular’ follows because F ∩ Q̃ ≅ K̸ ∩ Q̃ (one is real closed, the other not), so F ≢ K.□

5.3 The end

To conclude the proof of the negative part of the Main Theorem, let K be elementarily characterized by G_{K}, assume that K is not real closed and that either G_{K} is pro-solvable or that (there is a field elementarily equivalent to) K (which) is of finite transcendence degree over its prime field. We have to show that K is of type 2, 3 or 4.

By Corollary 5.5, the canonical henselian valuation v on K is then of mixed characteristic, say (0, p). If all henselian valuations with residue characteristic 0 have divisible value group then we may assume that O_{v}[\frac{1}{p}] = K, and so, by Proposition 3.2 and Theorem 3.3, that K is p-adically closed, hence of type 2.

If there is some henselian valuation w on K with char Kw = 0 and Γ_{w} ≠ qΓ_{w} for some prime q, then from Proposition 5.4 and from what has by now become standard trickery, we may assume Γ_{w} ≡ Z_{(q)} and Kw is elementarily determined by G_{Kw} and μ_{q^{∞}} among the fields of characteristic ≠ q and that O_{v/w}[\frac{1}{p}] = Kw.

If now cd_{p}G_{Kw} > 1, then, again Kw is of type 2 by 3.2 and 3.3: note that, for p-adically closed fields L, char L and μ_{L} is encoded in G_{L}; so K is of type 3.

If, finally, cd_{p}G_{Kw} ≤ 1, then, by Theorem 3.3 and Lemma 5.2, Kw has no proper abelian extension (hence μ_{p^{∞}} ⊆ Kw), so it is elementarily determined by G_{Kw} among all fields of characteristic ≠ p and tr.deg.K/Q < ∞, as G_{K} is then not pro-solvable. Moreover, cd_{p}G_{Kw} = 0 is impossible since, otherwise, G_{K} could be realized as absolute Galois group of a henselian field of characteristic p by the technique in [Ko5], Theorem 4.6 proof (c.1.2): By Theorem 3.3, G_{Kw} ≅ G_{F} for some field F of characteristic p. Let E := F ((t)) with power series valuation u and let α ∈ E^{sep} be a zero of the polynomial X^{p} + X + t^{−1}. Then E(α)/E is a (w.r.t. u) purely ramified Galois extension of degree p. Let E′ := ⋃_{(n,p)=1} E( ^{n} \sqrt{t}) and let K′ be a maximal separably algebraic p-extension not containing α. Then, by [G], Satz 4.9, the inertia subgroup V of G_{K′} (w.r.t. the unique prolongation of u) is ≅ Z_{p} and

G_{K′} ≅ Z_{p} × G_{F} ≅ Z_{p} × G_{Kw} ≅ G_{K}.

Hence K is of type 4, indeed.

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Universität Konstanz, Mathematik, Postfach 5560, D-78434 Konstanz, Germany

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