Fields Institute Communications Volume 00, 0000

Encoding valuations in absolute Galois groups

Jochen Koenigsmann

Universität Konstanz Fachbereich Mathematik und Statistik D-78457 Konstanz, Germany

                       Jochen.Koenigsmann@uni-konstanz.de

Abstract. The paper develops an Artin-Schreier theory for valued fields: mutatis mutandis, valuations on any field F are encoded in the absolute Galois group G_{F} of F, just like orderings. We present a Galois characterization of henselian fields as well as a purely group theoretic characterization of decomposition subgroups of G_{F} for arbitrary fields F. The approach is elementary and self-contained.

Contents

1. Introduction 2 1.1 The question 2 1.2 Notation 2 1.3 Main results 2 1.3.1 Tamely branching henselian fields 2 1.3.2 Abstract decomposition subgroups of G_{F} 3 1.3.3 Hensel equivalence 5 1.4 ‘Historical’ remarks 5 1.5 What’s new? 6 2. p-henselian fields 6 2.1 Basics 6 2.2 Comparability of p-henselian valuations on a fixed field 7 2.3 Canonical (p-)henselian valuations 9 2.4 Rigid elements and valuations 10 2.5 The wonderful creation of rigid elements 12 2.6 Galois characterization of p-henselian fields 14 2.6.1 The classical case 15 2.6.2 The general case 16

\begin{matrix}c \\ ◯ \end{matrix}0000 American Mathematical Society

1991 Mathematics Subject Classification. Primary 12F10, 12J10; Secondary 12G05.

Partly written while the author enjoyed the hospitality of MSRI, Berkeley, as research member in Fall 1999.

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

2.7 Maximal abelian normal subgroups of G_{F} (p) 17 3. Henselianity inherited from Sylow fields 17 4. Proofs of the main results 19 4.1 Galois characterization of henselian fields 19 4.2 Group theoretic characterization of decomposition subgroups of G_{F} 21 4.3 Examples 23 References 25

1 Introduction

1.1 The question. Artin-Schreier-theory for ordered fields may be viewed as a way of encoding orderings on a field F in its absolute Galois group G_{F} := Gal(F^{sep}/F): F is real closed iff ♯G_{F} = 2 iff 1 < ♯G_{F} < ∞; F is formally real (i.e., admits an ordering) iff G_{F} contains an involution (= an element of order 2); and the orderings on F are in canonical 1-1 correspondence to the conjugacy classes of involutions in G_{F}.

Given the long list of analogies between orderings and valuations, it is quite natural to ask for a valuation theoretic counterpart: are valuations on a field F encoded in G_{F}? In particular, is being henselian (a pendant to being real closed) a Galois-theoretic property? And, even asking for more, is there a Galois characterization of henselisations, i.e., a group theoretic characterization of decomposition subgroups of G_{F}? The answer is yes (with few exceptions).

1.2 Notation. For a valued field (F, v) we denote the valuation, the valuation ring, its maximal ideal, the residue field and the value group by v, O_{v}, M_{v}, F v, and Γ_{v} repectively. F^{sep} and F̃ are separable and algebraic closure of a field F. And G_{F} := Gal(F^{sep}/F) is the absolute Galois group of F, always considered as profinite group. In particular, all subgroups are assumed to be closed and all homomorphisms continuous.

We say that a valuation v on a field F is tamely branching if there is a prime p ≠ char F v with Γ_{v} ≠ pΓ_{v} and, if [Γ_{v} : pΓ_{v}] = p then p^{2} | ♯G_{F v} (as supernatural numbers, i.e., there is a finite separable extension L/F v with p^{2} | [L : F v]). If p is such a prime we say v is tamely branching at p. Note that the condition ‘p^{2} | ♯G_{F v}’ is equivalent to ‘p | ♯G_{F v}’, unless p = 2 and F v is formally real (cf. [31]).

1.3 Main results.

1.3.1 Tamely branching henselian fields.

Theorem 1 A field F admits a tamely branching henselian valuation iff G_{F} has a non-procyclic Sylow subgroup P ≇ ℤ_{2}⋊ℤ/2ℤ with a non-trivial abelian normal subgroup N ◁ P.

One direction of this Galois characterization of henselian fields is an immediate consequence of Hilbert ramification theory (see, e.g., [9], §19 and §20): if F admits a henselian valuation v which is tamely branching at p then the inertia subgroup of a Sylow-p-subgroup P of G_{F} is a non-trivial abelian normal subgroup of P and P is neither procyclic nor isomorphic to ℤ_{2} ⋊ ℤ/2ℤ.

Theorem 1 characterizes all tamely branching henselian valued fields in Galoistheoretic terms. But for henselian fields which are not tamely branching henselianity does not show in the absolute Galois group, as the following examples illustrate:

G𝔽̃_{p} =  1  = GQ̃
G_{𝔽_{p}}  ≅  ℤ̂  ≅ G_{ℂ((t))}
G_{ℝ}  ≅  ℤ/2ℤ  ≅ G_{ℝ^{⋆}}
G_{R∩σR} ≅ ℤ̂ ⋊ ℤ/2ℤ ≅ G_{ℝ((t))}
G_{K}

≅

Demuškin group of rank ℵ_{0}

≅
G_{^{p}ℚ_{p}}.

Here ℝ^{⋆} denotes a nonstandard extension of the reals (with the convex hull of ℤ as non-trivial henselian valuation ring), R is any real closure of ℚ, σ is any element ∈ G_{ℚ} with ⟨σ⟩ ≅ ℤ̂, ^{p}ℚ_{p} is the fixed field of a Sylow-p-subgroup of G_{ℚ_{p}} and K is the non-henselian field with G_{K} ≅ G_{^{p}ℚ_{p}} constructed in [34], Remark 2.6.

There are two key ingredients in the proof of Theorem 1 (which was first proved as Corollary 2.5 in [29]). One is a Galois-characterization of so-called p-henselian fields ([15] for p = 2 and [13] for p > 2). We give a new elementary proof of this characterization in section 2. The other ingredient is the observation that, if the fixed field of a non-trivial Sylow-subgroup of G_{F} admits a non-trivial henselian valuation, then—apart from one exception—so does F (Theorem 3.1).

1.3.2 Abstract decomposition subgroups of G_{F}. We have to face two specifically valuation theoretic phenomena where the analogy between henselian fields and real closed fields fails. One is that a henselian valuation (uniquely) extends to a henselian valuation on any algebraic field extension, while real closed fields are algebraically maximal w.r.t. their ordering. The other is that henselian valued fields may have different henselian valuations, whereas real closed fields are uniquely ordered.

So, given an arbitrary field F, we can use Theorem 1 only to describe the subgroups of G_{F} corresponding to separably algebraic extensions of F with tamely branching henselian valuations. However, it does not provide a criterion indicating which of these subgroups correspond to henselisations of G_{F} w.r.t. tamely branching valuations on F, i.e., it gives no group theoretic characterization of decomposition subgroups of G_{F}.

The first naive guess in this direction would be to try maximal subgroups of G_{F} of this type. And indeed, if F —like all global fields—admits only tamely branching valuations of (archimedean) rank 1, then the decomposition subgroups of G_{F} are exactly the maximal subgroups of G_{F} which have a non-procyclic Sylow subgroup P ≇ ℤ_{2} ⋊ ℤ/2ℤ with a non-trivial normal abelian subgroup N ◁ P.

The general case is slightly more involved. We have to confine our characterization to decomposition subgroups of G_{F} w.r.t. regular tamely branching valuations v on F: we call a tamely branching valuation v on F regular if Γ_{v} has no non-trivial convex subgroup which is p-divisible by all primes p at which v is tamely branching.

Passing to the quotient of Γ_{v} by a maximal convex subgroup of this type canonically associates to any tamely branching valuation v a regular (at the same primes) tamely branching coarsening of v. Of course, tamely branching rank-1 valuations are always regular.

We will see that, in general, there can be no group theoretic characterization of decomposition subgroups of G_{F} w.r.t. non-regular tamely branching valuations on F (Example 4.7).

Our characterization requires some more notation. Let F be a field and let p be a prime. For any pro-p subgroup P of G_{F} the (topological) closure in G_{F} of the union of an ascending chain of subgroups of G_{F} with P as Sylow-p-subgroup has, again, P as Sylow-p-subgroup. So there are (w.r.t. inclusion) maximal subgroups with this property and we may define

D(P) := {D ≤ G_{F} | D maximal with P ≤ D and p ∤ [D : P]}.

By Corollary 4.2, any pro-p subgroup P of G_{F} has a unique maximal normal abelian subgroup which we denote by N_{P} ◁ P. Still fixing a prime p, let

P_{p} := P_{p}(G_{F}) := {P ≤ G_{F} | P pro-p gp. ≇ ℤ_{p}, ℤ/2ℤ or ℤ_{2} ⋊ ℤ/2ℤ & N_{P} ≠ 1}

and define a partial order ‘⪯’ on P_{p}:

P, P′ ∈ P_{p} : P ⪯ P′ ⇔ P ≤ P′ and N_{P} = N_{P′} ∩ P.

Since, again, the closure in G_{F} of the union of an ascending chain of elements in P_{p} belongs to P_{p}, there are maximal elements in P_{p}. Write

P_{p,max} := {P ∈ P_{p} | P maximal w.r.t. ⪯},
P_{p,max}^{anab} := {P ∈ P_{p,max} | [P : N_{P}] > 2}.

Now define an abstract decomposition subgroup of G_{F} to be a subgroup D ≤ G_{F} such that

D = ⋂_{p∈S} D_{p},

where S is some non-empty set of rational primes, where

D_{p} ∈ ⋃_{P ∈P_{p,max}} D(P)

for each p ∈ S, and where, for any p, q ∈ S, D_{p} ⊆ D_{q} or D_{q} ⊆ D_{p}.

An abstract decomposition subgroup of G_{F} is called anabelian if D can be obtained as such an intersection D = ⋂_{p∈S} D_{p}, where, in addition, for each p ∈ S,

D_{p} ∈ ⋃_{P ∈P_{p,max}^{anab}} D(P).

Of course, any D ∈ D(P) for any P ∈ P_{p,max}^{anab} is an anabelian abstract decomposition subgroup of G_{F}.

Finally, we call a regular tamely branching valuation v on F anabelian if there is a non-empty set S of primes at which v tamely branches such that O_{v} = ⋂_{p∈S} O_{v_{p}}, where, for each p ∈ S, v_{p} is a non-trivial valuation on F such that O_{v_{p}} ⊇ O_{v}, such that Γ_{v_{p}} has no non-trivial p-divisible convex subgroup, and such that either G_{F v_{p}} has non-procyclic Sylow-p-subgroups ≇ ℤ_{2} ⋊ ℤ/2ℤ or the Sylowp-subgroups of G_{F v_{p}} are ≅ ℤ_{p} and μ_{p^{∞}} ⊈ F v(ζ_{p}), where ζ_{p} is a primitive p-th root of unity and μ_{p^{∞}} is the group of all p^{n}-th roots of unity (n ∈ ℕ), or p = 2, the Sylow-2-subgroups of G_{F v_{2}} are ≅ ℤ_{2} ⋊ ℤ/2ℤ and μ_{2^{∞}} ⊈ F v(\sqrt{−1}).

Theorem 2 Let F be a field. Then:

(a) Every decomposition subgroup of G_{F} w.r.t. a regular tamely branching valuation on F is an abstract decomposition subgroup of G_{F}.

(b) D ≤ G_{F} is an anabelian abstract decomposition subgroup of G_{F} iff D is a decomposition subgroup of G_{F} w.r.t. an anabelian regular tamely branching valuation v on F.

We will see that an abstract decomposition subgroup of G_{F} which is not anabelian may or may not be a decomposition subgroup of some valuation on F (Example 4.6).

As a borderline case, an anabelian abstract decomposition subgroup may happen to be abelian (as profinite group), thus forgetting its anabelian origin (Example 4.8).

1.3.3 Hensel equivalence. We call two valuations v, w on a field F Hensel equivalent if there is a valuation u on F with O_{v} ⊆ O_{u} and O_{w} ⊆ O_{u} such that the valuations v/u resp. w/u induced by v resp. w on F u are henselian. Equivalently, v and w are Hensel equivalent if they have the same henselisations. With this terminology, Theorem 2(b) immediately gives

Theorem 3 There is a 1-1 correspondence between Hensel equivalence classes of regular anabelian tamely branching valuations on a field F and conjugacy classes of anabelian abstract decomposition subgroups of G_{F}.

Note that the Hensel equivalence class of a regular anabelian tamely branching valuation on F may contain non-regular anabelian tamely branching valuations or valuations which are not tamely branching at all.

1.4 ‘Historical’ remarks. The Hasse-Brauer-Noether local-global-principle for the Brauer group of algebraic extensions of global fields may be considered as the first instance of detecting valuation theoretic information in the absolute Galois group of a field: by Galois-cohomology, the Brauer group of F is determined by G_{F} (Br(F) ≅ H^{2}(G_{F} , ℚ/ℤ)).

The wish to find a Galois characterization of the most classical henselian fields, the fields ℚ_{p} of p-adic numbers, or, more generally, of p-adically closed fields, seems to go back to Krull. The first result in this direction was Neukirch’s Galois characterization of ℚ_{p} ∩ ℚ̃ among the subfields of ℚ̃ ([36]), making substantial use of the local-global-principle for the Brauer group. Uchida ([46]) proved the analogue for function fields of curves over finite fields. Pop then discovered a relative localglobal-principle for the Brauer group of function fields over p-adically closed fields and thus arrived at a Galois characterization of p-adically closed fields among all fields F (of characteristic 0) with F̃ = F ℚ̃ ([37]). The full Galois characterization of p-adic fields was then obtained in [11] and in Pop’s Appendix to [26] for p ≠ 2 and, independently, in [26] for all p. It is no longer based on a local-global-principle for the Brauer group (it is well known that such a principle fails in general). However, it will become clear from the proof of our main results that it can be based on something which may be phrased in terms of another local-global-principle:

Local-global-principle for metabelian subgroups of G_{F} Let F be an arbitrary field. Then any torsion-free metabelian subgroup of G_{F} of rank > 1 is contained in G_{F^{v}} for some henselisation F^{v} of F w.r.t. some valuation v on F.

There have been various constructions of valuations in the context of topology (e.g., [23], [18], [40]), of geometry (e.g., [2], [8], [22], [24]) and of quadratic form theory (e.g., [39], [3], [6], [7], [20], [48], [1]). Most of them were phrased in some terms of rigid elements. The connection that these rigid elements can be seen in G_{F} was made in [48] for 2-rigid elements and in [26], [19] and [27] in general.

The first—and to my knowledge the only—group theoretic characterizations of decomposition subgroups beyond the global field (and the henselian) case are Bogomolov’s characterization of inertia subgroups (of a certain pro-p quotient of G_{F}) for ‘divisorial’ valuations on function fields in at least two variables over an algebraically closed field ([5], Lemma 4.1), Nakamura’s characterization of decomposition subgroups of arithmetic étale fundamental groups of punctured projective lines corresponding to the deleted points ([35], Theorem 3.4), and Pop’s Galois characterization of ‘defectless divisorial’ valuations on fields which are finitely generated over their prime field ([38], Theorem 1.16; a different characterization can be obtained from [44], Proposition 2.6). It is easy to see that all these valuations are anabelian regular tamely branching valuations. So our Theorem 2(b) may be considered as a generalisation to arbitrary fields and almost arbitrary valuations. Note that Tamagawa’s characterization of decomposition subgroups of arithmetic fundamental groups of affine hyperbolic curves over finite fields, p-adically closed fields and fields finitely generated over ℚ ([45], Corollary 2.10), can only be recovered from our Theorem when the absolute Galois group of the corresponding function field is recovered from the fundamental group.

1.5 What’s new? This paper was designed to reproduce my report on the ‘Galois characterization of henselian fields’ (Theorem 1) presented at the Valuation Theory Conference in Saskatoon in summer 1999. Meanwhile, however, we considerably simplified the original proof (Corollary 2.5 in [29]) by the observation that henselianity is inherited from ‘Sylow-fields’ (Theorem 3.1). This observation then made clear that the same valuation theoretic techniques as for the proof of Theorem 1 could be further exploited to develop a full general account of decomposition subgroups of absolute Galois groups. Since this problem could sofar only be solved in very special situations (see above), we thought it worthwhile to include our general solution in this paper. In [30], our general solution will be applied to a birational version of the so-called ‘section-conjecture’ in anabelian geometry over arbitrary fields.

Another novelty in our approach is the fact that it is completely elementary. In particular, no Galois cohomology is required (except, perhaps, for the concluding examples). So we also give a new elementary proof for the Galois characterization of p-henselian fields (Theorem 2.15) avoiding the tools from Galois cohomology and model theory involved in the original proofs ([15] and [13]).

2 p-henselian fields

2.1 Basics. For any field F and any prime p we denote by F (p) the maximal Galois-p extension of F, i.e., the compositum of all finite Galois extensions L/F in F^{sep} where ♯Gal(L/F) is a power of p. Then F (p)/F is a Galois extension with a pro-p Galois group denoted by G_{F} (p) := Gal(F (p)/F) ≅ G_{F} /G_{F (p)}, the maximal pro-p quotient of G_{F}.

Fact 2.1 (F (p))(p) = F (p).

Proof G_{F (p)} = ⋂_{N} N, where N runs through all open normal subgroups of G_{F} of p-power index. Thus, G_{F (p)} is a characteristic subgroup of G_{F}. Similarly, G_{(F (p))(p)} is a characteristic subgroup of G_{F (p)}, so, in particular, it is fixed under automorphisms of G_{F (p)} induced by conjugation with elements from G_{F}. Therefore, G_{(F (p))(p)} is a normal subgroup of G_{F} with pro-p quotient. As G_{F} (p) is the maximal pro-p quotient of G_{F} this implies G_{(F (p))(p)} = G_{F (p)}, and hence (F (p))(p) = F (p).

By Artin-Schreier-Becker-theory ([3], Satz 3), the only non-trivial torsion elements of G_{F} (p) are involutions, i.e., elements of order 2, and their corresponding fixed fields are Euclidean, i.e., the squares are the positive elements of an ordering. Of course, this can only happen for p = 2.

A valuation v on F is called p-henselian if v extends uniquely to F (p) or, equivalently, if Hensel’s Lemma holds for polynomials splitting into linear factors over F (p). Indeed, one only has to focus on Galois extensions of degree p:

Fact 2.2 ([47], Proposition 1.1) (F, v) is p-henselian iff v extends uniquely to every Galois extension of F of degree p.

Proof Assume F extends uniquely to every Galois extension of degree p. Let (L, w)/(F, v) be a finite Galois extension with L ⊆ F (p). Let D be the decomposition subgroup of G := Gal(L/F) w.r.t. w.

If D ≠ G, i.e., if v does not extend uniquely to L, we find a subnormal series for the finite p-group G through D, and, in particular, some normal subgroup N ◁ G with D ≤ N and [G : N] = p. Under the canonical projection G → G/N, D projects onto the decomposition subgroup of G/N (w.r.t. the restriction of w to the fixed field of N). Yet the latter coincides, by assumption, with G/N, i.e., N = DN = G ≠ N: contradiction.

So v extends uniquely to any finite Galois subextension of F (p)/F and hence to F (p).

The converse is trivial.

By Kummer theory, the fact together with p-Hensel’s Lemma readily implies:

Corollary 2.3 Let (F, v) be a valued field with char F v ≠ p and assume F contains a primitive p-th root ζ_{p} of unity.

Then v is p-henselian iff 1 + M_{v} ⊆ F^{p}.

Examples 2.4 1. Every henselian valuation is p-henselian.

2. If (F, v) is any valued field with prolongation w to F (p) then the fixed field of the decomposition subgroup of G_{F} (p) w.r.t. w is p-henselian, but, in general, not henselian: e.g., if F = ℚ and v is any non-trivial valuation on F.

2.2 Comparability of p-henselian valuations on a fixed field. Recall that two valuations v, w on a field F are called comparable if O_{v} ⊆ O_{w} (‘v is finer than w’) or O_{w} ⊆ O_{v} (‘v is coarser than w’). One easily verifies the equivalences

O_{v} ⊆ O_{w} ⟺ O_{v}^{×} ⊆ O_{w}^{×} ⟺^{!} M_{w} ⊆ M_{v}.

If v is finer than w then v induces a valuation v/w on the residue field F w of w and there is an exact sequence of ordered abelian groups

0 → Γ_{v/w} → Γ_{v} → Γ_{w} → 0.

By Hensel’s Lemma (or the p-henselian analogue) v is then henselian (resp. phenselian) iff both w and v/w are henselian (resp. p-henselian).

Generalizing the theorem of F. K. Schmidt that a non-separably closed field admits at most one henselian rank-1 valuation, Engler and Endler show in [EE] that on any field two henselian valuations with non-separably closed residue field are comparable, that, if there is any, then there is a coarsest valuation among the henselian valuations with separably closed residue field and that each of them is finer than any henselian valuation with non-separarbly closed residue field.

In order to prove the p-henselian analogue of these results we need the following slight generalisation of the Proposition in [16]:

Lemma 2.5 Let (F, v) be a valued field and let E/F v be a finite abelian Galois extension such that 8 ∤ exponent of A := Gal(E/F v). Then there is a purely inert Galois extension (L, w)/(F, v) with Lw = E; so, in particular, Gal(L/F) ≅ Gal(E/F v) = A.

Proof We may assume A to be cyclic of order p^{n} for some prime p and some n ∈ ℕ, where n ≤ 2 for p = 2.

Then, by [41] and [43], there is a generic polynomial for A-extensions outside characteristic p, i.e., there is a monic polynomial g ∈ ℤ[t_{1}, …, t_{r}, X] such that for any field K with char K ≠ p and for any a_{1}, …, a_{r} ∈ K, the polynomial g(a_{1}, …, a_{r}, X)‘∈’K[X] (if char K = q > 0 consider g as polynomial in 𝔽_{q}[t_{1}, …, t_{r}, X]) either factors or it is the irreducible plynomial of a primitive element of an A-extension of K (i.e., a Galois extension of K with group A) and any A-extension of K is obtained by adjoining a root of g after a suitable specialisation t_{i} ↦ a_{i} ∈ K (i = 1, …, r).

So, if char F v ≠ p we can choose a_{1}, …, a_{r} ∈ O_{v} such that g(\overline{a}_{1}, …, \overline{a}_{r}, X) ∈ F v[X] is the irreducible polynomial for a primitive element for E/F v. Then g(a_{1}, …, a_{r}, X) ∈ F [X] is irreducible as well, and, thus by genericity, defines an Aextension L/F. Inducing the A-extension E/F v over the residue field, the extension L/F must be purely inert.

On the other hand, if char F = p (and so also char F v = p), we can repeat the argument using the fact that in characteristic p there are again generic polynomials for A-extensions: if n = 1 one has the Artin-Schreier polynomial X^{p} + X + t ∈ 𝔽_{p}[t, X], for arbitrary n see, e.g., [33], III.4.3.

Hence, we are left with the mixed characteristic case: char F = 0 and char F v = p. In this case we may assume v to be of (archimedean) rank 1: This is because v admits a coarsest coarsening v_{p} with char F v_{p} = p and a finest coarsening v_{0} with char F v_{0} = 0. Then v_{p} is finer than v_{0} and induces a rank-1 valuation v_{p}/v_{0} on F v_{0} of mixed characteristic (0, p). By the considerations above, A-extensions lift from F v to F v_{p} and from F v_{0} to F, so it remains to lift them from F v_{p} to F v_{0}.

As v is now a rank-1 valuation, (K, v) is dense in its henselisation(F̂, v̂) (or, equivalently, the completion of (F, v) is henselian). Let T be the inertia subgroup of GF̂ and let π be the compositum of the epimorphisms

GF̂ ↠ GF̂/T ≅ G_{F̂v̂} ≅ G_{F v} ↠ Gal(E/F v).

Then the fixed field of the kernel of π is a purely inert A-extension lifting E/F v. Now approximate the irreducible polynomial of a primitive element for this extension obtained from suitable substitutions of a generic polynomial for A-extensions in F̂ by a polynomial over F to find the required lifting of E/F v over F.

Corollary 2.6 Let (F, v) be a valued field with F = F (p). Then F v = F v(p).

Corollary 2.7 If a field F admits two independent p-henselian valuations, then F = F (p).

Proof Let g(t_{1}, …, t_{r}, X) ∈ F [t_{1}, …, t_{r}, X] be a generic polynomial for ℤ/pℤextensions over F. Assume v, w are independent p-henselian valuations and F ≠ F (p). Then there are a, b ∈ F^{r} such that adjoining a zero of g(a, X) to F gives a Galois extension of degree p, whereas g(b, X) has a zero in F. Now approximate a w.r.t v and b w.r.t. w well enough by some c ∈ F^{r} to guarantee (with Krasner’s Lemma) that g(c, X) generates the same extension in the henselisation of (F, v) as g(a, X) and that g(c, X) has a zero in a henselisation of (F, w). Then g(c, X) is irreducible and, by genericity, thus generates a ℤ/pℤ-extension of F. In particular, g(c, X) splits in linear factors over F (p). Hence, by the p-henselian variant of Krasner’s Lemma w.r.t. w, g(c, X) must have a zero in F: contradiction.

Proposition 2.8 Let F be a field, let p be a prime and define

H_{1} := {v | v p-hens. val. with F v ≠ F v(p)},
H_{2} := {v | v p-hens. val. with F v = F v(p)}.

Then any two valuations in H_{1} are comparable, and, if H_{2} ≠ ∅, then H_{2} contains a valuation which is coarser than any other valuation in H_{2} and strictly finer than any valuation in H_{1}. If H_{2} = ∅, there is a finest valuation in H_{1}.

Proof If v and w are incomparable p-henselian valuations on F then they induce independent p-henselian valuations v/u and w/u on the residue field F u of their finest common coarsening u on F (O_{u} = O_{v}O_{w}). Hence, by the previous Corollary, F u = F u(p), and so, by Corollary 2.6,

F v = (F u)(v/u) = (F u)(v/u)(p) = (F v)(p),

i.e., v ∈ H_{2}, and, similarly, w ∈ H_{2}.

If v ∈ H_{2} and w ∈ H_{1}, then v is strictly finer than w: we just saw that v and w are comparable, but that v is finer than w is impossible, again by Corollary 2.6 applied to (F w, v/w).

For any v, w ∈ H_{2} their finest common coarsening u again belongs to H_{2}: v/u and w/u are independent p-henselian valuations on F u, so F u = F u(p) by Corollary 2.7. Thus, for any v ∈ H_{2},

O := ⋃_{w∈H_{2}} O_{w} = ⋃_{v≤w∈H_{2}} O_{w},

where ‘≤’ means ‘finer than’. So O = O_{v_{O}} is the limit (union) of a linear chain of valuation rings O_{w} with F w = F w(p). Hence F v_{O} = F v_{O}(p), i.e., v_{O} ∈ H_{2}.

Since the v ∈ H_{1} are pairwise comparable, ⋂_{v∈H_{1}} O_{v} is again a valuation ring, corresponding to a p-henselian valuation. If H_{2} = ∅, it is the finest valuation in H_{1}.

2.3 Canonical (p-)henselian valuations. The last proposition resp. [10], justifies the following

Definition 2.9 The canonical p-henselian resp. henselian valuation v on a field F is the coarsest p-henselian resp. henselian valuation on F with F v = F v(p) resp. F v = F v^{sep}, if there is any. Otherwise v is the finest p-henselian resp. henselian valuation on F.

It is clear that the canonical (p-)henselian valuation is non-trivial iff there is a non-trivial (p-)henselian valuation on F and F ≠ F (p) resp. F ≠ F^{sep}. Any valuation coarser than the canonical (p-)henselian valuation is again (p-)henselian and is comparable with any other (p-)henselian valuation on F.

The following Proposition comprises [14], Cor. 3.5, [38], Prop. 1.3 and [13], Lemma 3.2. Note that in Pop’s terminology, a valuation is a coarsening of the canonical henselian valuation on F iff it ‘equals its F -core’. There is no connection, however, between our canonical p-henselian valuations and Pop’s notion of a ‘p, F -core’ of a henselian valuation on F.

Proposition 2.10 Let F be a field with F ≠ F (p) resp. F ≠ F^{sep} and let L/F be any normal or finite proper subextension of F (p)/F resp. F^{sep}/F. Then the restriction of any coarsening of the canonical (p-)henselian valuation on L to F is a coarsening of the canonical (p-)henselian valuation on F; in particular, it is (p-)henselian again.

Proof If L/F is Galois and v is a coarsening of the canonical (p-)henselian valuation on L then any conjugate of v over F is p-henselian again. But distinct conjugates are incomparable, so v is the only prolongation of v | F to L, i.e., v | F is again (p-)henselian.

Since v is a coarsening of the canoncial (p-)henselian valuation, any proper coarsening v′ of v satisfies Lv′ ≠ Lv′(p) resp. Lv′ ≠ Lv^{′sep}, which implies F v′ ≠ F v′(p) resp. F v′ ≠ F v^{′sep}. As the proper coarsenings of v | F are induced by those on L this shows that v | F is a coarsening of the canonical (p-)henselian valuation on F.

If L/F is finite and v is a coarsening of the canonical (p-)henselian valuation on F, then the normal hull of L/F is a proper Galois subextension of F (p)/F resp. F^{sep}/F (recall that the only non-trivial torsion elements in G_{F} (p) resp. G_{F} have order 2). Let w be the unique (again (p-)henselian) prolongation of v to N and let w′ be the canoncial (p-)henselian valuation on N.

If w′ is finer than w then, by what we have already proved, u := w | F = v | F is a coarsening of the canonical (p-)henselian valuation on F.

If w′ is not finer than w then w′ is a proper coarsening of w and F w′ = F w′(p) resp. F w′ = F w^{′sep}. But Lv′ ≠ Lv′(p) resp. Lv′ ≠ Lv^{′sep} for v′ := w′ | L, and hence F u′ ≠ F u′(p) resp. F u′ ≠ F u^{′sep} for u′ := w′ | F = v′ | F. Thus, by Artin-Schreier-Becker-theory, p = 2 and both Lv′ and F u′ are Euclidean resp. real closed. So Lv′ = F u′ and Nw′ = Lv′(\sqrt{−1}). As v is (p-)henselian on L, the induced valuation v/v′ = u/u′ is (p-)henselian on Lv′ = F u′. Moreover, since N/F is Galois, u′ is (p-)henselian on F, and so u is (p-)henselian. Since F u′ is Euclidean resp. real closed the same applies to F u = (F u′)(u/u′). Hence u is a coarsening of the canonical (p-)henselian valuation on F.

2.4 Rigid elements and valuations. The ultrametric triangle inequality for a valuation v on a field F implies that for any x ∈ F^{×} \ O_{v}^{×}:

O_{v}^{×} + x · O_{v}^{×} ⊆ O_{v}^{×} ∪ x · O_{v}^{×}

The same ‘rigidity’ holds when O_{v}^{×} is replaced by any subgroup T ≤ F^{×} with O_{v}^{×} ≤ T.

It turns out that, conversely, subgroups of F^{×} (of index ≠ 2 or 4) with this property give rise to valuations. For simplicity, we shall prove this only in case (F^{×})^{p} ⊆ T for some prime p > 2 (Proposition 2.13). The general case is treated in [1], extending earlier result from [20] and [48] and proceeding along the same elementary lines.

Definition 2.11 Given a subgroup T ≤ F^{×}, define

O_{1}(T) := {x ∈ F \ T | 1 + x ∈ T },
O_{2}(T) := {x ∈ T | xO_{1}(T) ⊆ O_{1}(T)},
O(T) := O_{1}(T) ∪ O_{2}(T).

An element x ∈ F^{×} \ T is called T -rigid if T + xT ⊆ T ∪ xT.

Examples 2.12 1. If T = F^{×} then O(T) = F.

2. If F = ℂ((t)) with the power series valuation v and if T = (F^{×})^{p} = v^{−1}(pΓ_{v}) then any x ∈ F^{×} \ T is T -rigid and O(T) = O_{v}:

O_{1}(T) = M_{v} \ (F^{×})^{p},
O_{2}(T) = O_{v} ∩ (F^{×})^{p} = O_{v}^{×} ∪ (M_{v} ∩ (F^{×})^{p}).

Proposition 2.13 Let p be a prime > 2. Assume that F is a field with a multiplicative subgroup T ≤ F^{×} such that (F^{×})^{p} ⊆ T and such that all elements in F^{×} \ T are T -rigid. Then O(T) is a valuation ring with O(T)^{×} ⊆ T.

Proof We write O, O_{1} and O_{2} for O(T), O_{1}(T) and O_{2}(T). Clearly, 0 ∈ O_{1} and 1 ∈ O_{2}. If O_{1} = {0} then O = F and we are done. So, from now on, O_{1} ≠ {0}. We will tacitly use the assumption that all elements in F^{×}\T are T -rigid throughout the proof again and again. Claim 1: −O_{1} = O_{1}.

x ∈ O_{1} ⟹^{p>2} −x ∉ T = −T & ± x^{2} ∉ T
⟹ 1 − x^{2} = (1 + x)(1 − x) ∈ (T ∪ x^{2}T) ∩ (T ∪ xT) = T
⟹ 1 − x ∈ T
⟹ −x ∈ O_{1}.

Claim 2: O · O ⊆ O. By definition, O_{2} ·O_{1} ⊆ O_{1} and O_{2} ·O_{2} ⊆ O_{2}, so we have to show that O_{1} ·O_{1} ⊆ O. Pick x, y ∈ O_{1}. Case 1: xy ∉ T.

⇒ 1 + xy = (1 + x) − x(1 − y) ∈ (T ∪ xyT) ∩ (T ∪ xT) = T
⇒ xy ∈ O_{1}.

Case 2: xy ∈ T. Pick any z ∈ O_{1} \ {0}. Then either xz ∉ T or yz ∉ T, say xz ∉ T.

⇒ xz ∈ O_{1} (case 1)
⇒ 1 + xyz = 1 − y + y(1 + xz) ∈ (T ∪ xyzT) ∩ (T ∪ yT) = T
⇒ xyz ∈ O_{1}.

Hence xy ∈ O_{2}. Claim 3: ∀x ∈ F : x ∈ O or x^{−1} ∈ O. For any x ∈ F^{×} \ T:

x ∈ O_{1} ⇔ 1 + x = x(1 + x^{−1}) ∈ T
⇔ 1 + x^{−1} ∈ x^{−1}T
⇔ x^{−1} ∉ O_{1}.

For any x ∈ T:

x ∉ O_{2} ⇔ ∃y ∈ O_{1} with xy ∉ O_{1}

⇒ (xy)^{−1} ∈ O_{1}

⇒^{2} x^{−1} = (xy)^{−1}y ∈ O_{1} · O_{1} ⊆ O.

Claim 4: 1 + O_{1} ⊆ O_{2}. For any x, y ∈ O_{1}:

xy ∉ T ⇒ 1 + (1 + x)y = (1 + y) + xy ∈ (T ∪ yT) ∩ (T ∪ xyT) = T
xy ∈ T ⇒ 1 + (1 + x)y = (1 + x)(1 + y) − x ∈ (T ∪ yT) ∩ (T ∪ xT) = T.

Hence (1 + x)y ∈ O_{1} and so 1 + x ∈ O_{2}. Claim 5: 1 + O_{2} ⊆ O. Pick x ∈ O_{2}. Case 1: 1 + x ∉ T.

⇒ −(1 + x) ∉ T & 1 − (1 + x) = −x ∈ T

⇒ −(1 + x) ∈ O_{1}

⇒^{1} 1 + x ∈ O_{1}.

Case 2: 1 + x ∈ T. Choose any y ∈ O_{1}.

⇒ \frac{−y}{1+y} ∉ T & 1 + \frac{−y}{1+y} = \frac{1}{1+y} ∈ T
⇒ \frac{−y}{1+y} ∈ O_{1}
⇒ 1 + (1 + x)y = (1 + y)(1 + (−x)\frac{−y}{1+y}) ∈ T
⇒ (1 + x)y ∈ O_{1}.

Hence 1 + x ∈ O_{2} Claim 6: O + O ⊆ O.

x, y ∈ O ⟹^{3} \frac{x}{y} ∈ O or \frac{y}{x} ∈ O, say \frac{x}{y} ∈ O

⟹^{2,4,5} x + y = y(1 + \frac{x}{y}) ∈ O.

Claim 1–6 show that O is a valuation ring of F. Moreover, the argument in claim 3 shows that O^{×} ⊆ O_{2} ⊆ T.

Note that the value group of the valuation corresponding to O(T) in the proposition has non-p-divisible value group, unless T = F^{×}. Using model theory, a much stronger result was obtained in [26] for the case that T = (F^{×})^{p} and ζ_{p} ∈ F: in order to find a valution with non-p-divisible value group one need not require that all x ∈ F \T are T -rigid, but only that there is one x ∈ F \T such that x, x^{2}, …, x^{p−1} are T -rigid. This result can also be generalized to the case (F^{×})^{p} ≤ T.

2.5 The wonderful creation of rigid elements. To keep the treatment selfcontained, we reproduce here the proof of [27], Lemma 3.3 which generalizes [26], Proposition 3.1, and which is also implicit in [19], 2.4–2.6. It gives a criterion for the existence of rigid elements. As the heading indicates, we still don’t have a good understanding why the criterion works. There should be some K-theory behind it, yet, sofar, it could not really be traced. Of course, trivial K-theoretic reformulations as in [12] are possible. There may also be some connection to Theorem 3.1 of [5].

Lemma 2.14 Let p be a prime > 2, let F be a field and let H and G be subgroups of F^{×} with (F^{×})^{p} ≤ H ≤ G, [G : H] ≥ p^{2} satisfying the condition

(⋆) ∀c ∈ G \ H : H + cH ⊆ ⋃_{i=0}^{p−1} c^{i}H.

Then there is a subgroup T of G containing H with [T : H] ≤ p such that

∀c ∈ G \ T : H + cH ⊆ H ∪ cH,

and hence also

∀c ∈ G \ T : T + cT ⊆ T ∪ cT.

Proof Let us assume that the conclusion of the lemma is false. Then we find elements a, b ∈ G with [⟨a, b⟩H : H] = p^{2}, and elements x, y ∈ H such that for some k, l ∈ {2, 3, …, p − 1}:

x + a ∈ a^{k}H,
y + b ∈ b^{l}H.

Claim: y − b ∈ b^{\frac{l}{2l−1}} H and 2l − 1 ≠ 0 ∈ 𝔽_{p}.

As (F^{×})^{p} ≤ H, we only have to consider exponents modulo p, i.e., we understand them as elements of 𝔽_{p}.

Using the hypothesis (⋆), we see that for some j, r, s, t ∈ 𝔽_{p},

xy + bx + ay = \begin{cases}xy + a(y + a^{−1}bx) & ∈ (a^{1−j}b^{j})^{r}H \\ b(x + b^{−1}(x + a)y) & ∈ b(a^{k}b^{−1})^{s}H \\ a(y + a^{−1}(y + b)x) & ∈ a(a^{−1}b^{l})^{t}H \end{cases}

So comparing the exponents of a and b yields

(1) r(1 − j) = ks = 1 − t
(2) rj = 1 − s = lt

As l ≠ 0, this implies r ≠ 0.

(1) + (2) ⇒ r = (l − 1)t + 1
(1) + k(2) ⇒ k = (kl − 1)t + 1

Therefore, as k ≠ 1, we have kl − 1 ≠ 0. Thus

t = \frac{k − 1}{kl − 1} and r = \frac{2kl − k − l}{kl − 1} ,

and so 2kl − k − l ≠ 0. ¿From (2) we then obtain

j = \frac{lt}{r} = \frac{l(k − 1)}{2kl − k − l} ≠ 0.

Now let y − b = b^{i}H. We have to compute i. We recall that bx + ay ∈ a^{1−j}b^{j}H. On the other hand

bx + ay = (b − y)x + (a + x)y ∈ a^{k}(a^{−k}b^{i})^{q}H

for some q. So 1 − j = k(1 − q) and j = iq = \frac{i(j+k−1)}{k} ≠ 0. As

j + k − 1 = \frac{(k − 1)(l + 2kl − k − l)}{2kl − k − l} = \frac{(k − 1)k(2l − 1)}{2kl − k − l},

this implies 2l − 1 ≠ 0 and

i = \frac{jk}{j + k − 1} = \frac{kl(k − 1)}{k(k − 1)(2l − 1)} = \frac{l}{2l − 1}

which proves the claim.

Note that it also follows that i ≠ 0, 1 as otherwise l = 0 or l = 1. Therefore, the set

I := {i ∈ 𝔽_{p} \ {0, 1} | ∃b′ ∈ ⟨b⟩H \ H ∃y ∈ H : y + b′ ∈ b^{′i}H}

is non-empty. Also, I is closed under i ↦ \frac{1}{i}: if y + b′ = b^{′i}z for some z ∈ H, then b′′ := −b^{′i}z satisfies y + b′′ ∈ (b′′)^{\frac{1}{i}} H. And, by the proof of the claim, I is closed under i ↦ \frac{i}{2i−1} and \frac{1}{2} ∉ I.

Such a set, however, does not exist. For, inductively, I contains no element of the form \frac{m}{m+1} for 1 ≤ m ≤ p − 2: \frac{1}{1+1} ∉ I and if \frac{m}{m+1} ∈ I and m > 1, then also

\frac{m − 1}{m} = (\frac{m}{m − 1})^{−1} = (\frac{\frac{m}{m+1}}{2\frac{m}{m+1} − 1})^{−1} ∈ I.

On the other hand, each i ∈ 𝔽_{p} can be written in this form:

i = \frac{\frac{i}{1−i}}{\frac{i}{1−i} + 1}

contradicting I ≠ ∅.

It follows that T + cT ⊆ T ∪ cT for any c ∈ G \ T, because T = ⋃_{i=0}^{p−1} d^{i}H for some d ∈ T, so given c ∈ G \ T, x, y ∈ H and i, j < p, one has cd^{i−j} ∈ G \ T and

(d^{i}x) + c(d^{j}y) = d^{i}(x + (cd^{j−i})y) ∈ d^{i}(H ∪ cd^{j−i}H) ⊆ T ∪ cT.

2.6 Galois characterization of p-henselian fields. By now we are prepared for the theorem which is one of the two main inputs for the Galois characterization of henselian fields: For p = 2 the theorem appears in section 4 of [15]. For p > 2 it is the Main Theorem of [13]:

Theorem 2.15 Let p be a prime, let F be a field containing a primitive pth root ζ_{p} of unity (so char F ≠ p) and assume G_{F} (p) ≇ ℤ_{p} and, if p = 2, also G_{F} (p) ≇ ℤ/2ℤ or ℤ_{2} ⋊ ℤ/2ℤ.

Then F admits a p-henselian valuation tamely branching at p iff G_{F} (p) has a non-trivial abelian normal subgroup.

As in Theorem 1, one direction is trivial: if F admits a p-henselian valuation v with Γ_{v} ≠ pΓ_{v} and char F v ≠ p then the inertia subgroup of G_{F} (p) is non-trivial (Γ_{v} ≠ pΓ_{v}), abelian (char F v ≠ p) and normal (v p-henselian, i.e., G_{F} (p) is its own decomposition subgroup w.r.t. v).

It is also clear that one has to exclude the exceptional cases G_{F} (p) ≅ ℤ_{p}, ℤ/2ℤ or ℤ_{2} ⋊ ℤ/2Z: each of them occurs among fields with and without any non-trivial p-henselian valuation:

\begin{aligned}G_{ℂ((t))}(p) & ≅ & ℤ_{p} & ≅ & G_{𝔽_{q}}(p) \\ G_{ℝ((ℚ))}(2) & ≅ & ℤ/2ℤ & ≅ & G_{ℝ}(2) \\ G_{ℝ((t))}(2) & ≅ & ℤ_{2} ⋊ ℤ/2ℤ & ≅ & G_{F} (2), \end{aligned}

where ℝ((ℚ)) denotes the field of formal Laurent series with coefficients in ℝ and exponents in ℚ, and where the field F is a subfield of ℚ̃ which is algebraically maximal w.r.t. two distinct orderings ([4]): the finite field 𝔽_{q} admits no non-trivial valuation at all, and fields with an archimedean ordering (like ℝ and F) admit no non-trivial 2-henselian valuation (it would have to be compatible with the ordering, contradicting archimedeanity).

For simplicity, again, we shall prove Theorem 2.15 only for odd primes.

2.6.1 The classical case. The classical case is when G_{F} (p) ≅ ℤ_{p} ⋊ ℤ_{p} (we do not specify the action here). This case is ‘classical’ because it covers the nonarchimedean local fields: if l is a prime ≠ p and if F is a finite extension of ℚ_{l} or of 𝔽_{l}((t)) containing ζ_{p} then G_{F} (p) ≅ ℤ_{p} ⋊ ℤ_{p}.

Lemma 2.16 Let p be a prime > 2 and let F be a field containing ζ_{p} with G_{F} (p) ≅ ℤ_{p} ⋊ ℤ_{p}. Then

(a) rk G_{L}(p) = dim_{𝔽_{p}} L^{×}/(L^{×})^{p} = 2 for any finite subextension L/F of F (p)/F.

(b) A bijection between (abelian) Galois extensions of degree p and subgroups of F^{×} of index p is established by the following ‘local reciprocity law’:

∀x ∈ F^{×} \ (F^{×})^{p} : N(x) = ⟨x⟩(F^{×})^{p},

where N(x) denotes the image of the norm (F (\sqrt[p]{x}))^{×} → F^{×}.

Proof (a) Since open subgroups of ℤ_{p} are isomorphic to ℤ_{p}, the split exact sequence

0 → ℤ_{p} → G → ℤ_{p} → 0

for G := ℤ_{p} ⋊ ℤ_{p} induces a sequence of the same shape for any open subgroup H of G. So H ≅ ℤ_{p} ⋊ ℤ_{p}, and, in particular, rk H = 2 = dim_{𝔽_{p}} Hom(H, 𝔽_{p}).

If L/F is a finite subextension of F (p)/F apply this to G = G_{F} (p) and H = G_{L}(p) and recall the Kummer isomorphism L^{×}/(L^{×})^{p} → Hom(G_{L}(p), ⟨ζ_{p}⟩) of 𝔽_{p}- vector spaces (ζ_{p} ∈ L).

(b) If L/F is a Galois extension of degree p, say L = F (\sqrt[p]{x}) for some x ∈ F \F^{p}, and if N_{L/F} denotes the corresponding norm then N_{L/F} (\sqrt[p]{x}) = x. Choose y ∈ F^{×} \ ⟨x⟩(F^{×})^{p}. Then, again by Kummer theory, y ∉ L^{p} and, since N_{L/F} (y) ∈ F^{p}, even y ∉ ⟨\sqrt[p]{x}⟩(L^{×})^{p}. As N_{L/F} (\sqrt[p]{x}) ∉ F^{p}, we also have \sqrt[p]{x} ∉ L^{p}, and hence, by (a), L^{×} = ⟨\sqrt[p]{x}⟩⟨y⟩(L^{×})^{p}. So N(x) = N_{L/F} (L^{×}) = ⟨x⟩(F^{×})^{p}.

Proof of Theorem 2.15 in the classical case Assume that p is a prime > 2, that F is a field containing ζ_{p} and that G_{F} (p) ≅ ℤ_{p} ⋊ ℤ_{p}.

Then, for H := (F^{×})^{p}, any x ∈ F^{×} \ H satisfies condition (⋆) of Lemma 2.14:

H + xH ⊆ ⋃_{i=0}^{p−1} x^{i}H.

This follows from the local reciprocity law of the previous lemma, since for a, b ∈ F^{×}

a^{p} + xb^{p} = N_{L/F} (a + b\sqrt[p]{x}) ∈ N(x) = ⟨x⟩(F^{×})^{p},

where L = F (\sqrt[p]{x}).

Hence, by Lemma 2.14, there is a proper subgroup T ≤ F^{×} with (F^{×})^{p} ≤ T such that any x ∈ F^{×} \ T is T -rigid. Proposition 2.13 then implies that F admits a valuation v with O_{v}^{×} ⊆ T. In particular, Γ_{v} ≠ pΓ_{v}.

This remains true when we pass to the coarsening of v obtained by dividing Γ_{v} by its maximal p-divisible convex subgroup. We may, thus, assume that Γ_{v} has no non-trivial p-divisible convex subgroup.

Now assume v is not p-henselian. Then there is a Galois extension L/F of degree [L : F] = p admitting several (hence exactly p immediate) prolongations v_{1}, …, v_{p} of v. By weak approximation, we find for each i ∈ {1, …, p} an element x_{i} ∈ M_{v_{i}} ∩ ⋂_{j≠i} O_{v_{i}}^{×}. Replacing, if necessary, x_{i} by x_{i} + a for some a ∈ M_{v} with v_{i}(x_{i}) > v_{i}(a) = v(a) ∉ pΓ_{v} (Γ_{v} has no non-trivial p-divisible convex subgroup) we may also assume v_{i}(x_{i}) ∉ pΓ_{v_{i}}. But then x_{1}, …, x_{p} are 𝔽_{p}-independent modulo (L^{×})^{p}, contradicting Lemma 2.16(a) (p > 2). Thus, v is p-henselian.

Finally, assume char F v = p. Since ζ_{p} ∈ F this implies char F = 0. As v(1 − ζ_{p}) = \frac{1}{p−1}v(p) > 0 and as Γ_{v} has no non-trivial convex p-divisible subgroup, we find some π ∈ F with 0 < v(π) < v(π^{2}) ≤ v(p) and v(π) ∉ pΓ_{v}. Then π, 1 + π and 1 + π^{2} are 𝔽_{p}-independent modulo (F^{×})^{p}: If x = π^{r}(1 + π)^{s}(1 + π^{2})^{t} ∈ (F^{×})^{p} for some r, s, t ∈ {0, …, p − 1} then r = 0 since v(π) ∉ pΓ_{v}, and so

x = (1 + π)^{s}(1 + π^{2})^{t} = (1 + y)^{p} = 1 + py + ⋯ + py^{p−1} + y^{p}

for some y ∈ M_{v}. If s ≠ 0 resp. s = 0 ≠ t then v(x − 1) = v(π) resp. v(π^{2}), so in any case v(x − 1) < v(py), and hence v(x − 1) = v(y^{p}) contradicting v(π) ∉ pΓ_{v}. Thus r = s = t = 0.

But Lemma 2.16(a) says that dim_{𝔽_{p}} F^{×}/(F^{×})^{p} = 2, so the assumption char F v = p was false and the classical case is proved.

2.6.2 The general case. In order to prove the general case of the non-trivial direction of Theorem 2.15 for primes p > 2, let F be a field containing ζ_{p} and assume that G_{F} (p) has a non-trivial abelian normal subgroup N. Let L be the fixed field of N.

If rk N > 1, then N ≅ ℤ_{p}^{α} for some α > 1 and there is a subgroup H ≤ N with H ≅ ℤ_{p} × ℤ_{p}. Let E be the fixed field of H. Then, by the classical case, E admits a p-henselian valuation v with Γ_{v} ≠ pΓ_{v} and char Ev ≠ p, and we may assume that v is a coarsening of the canonical p-henselian valuation. Therefore, by Proposition 2.10, v | L is again p-henselian (H ◁ N) and it is a coarsening of the canonical p-henselian valuation on L. Repeating the argument for the Galois extension L/F, we find v | F again p-henselian and, of course, still with non-p-divisible value group and char F v ≠ p.

Now assume rk N = 1, i.e., N ≅ ℤ_{p}. Since G_{F} (p) ≇ ℤ_{p}, we find some σ ∈ G_{F} (p) \ N such that

H := ⟨N, σ⟩ = N ⋊ ⟨σ⟩ ≅ ℤ_{p} ⋊ ℤ_{p} ≤ G_{F} (p).

Denote the fixed field of H by E again and apply, once more, the classical case to obtain a (p-henselian) coarsening v of the canonical p-henselian valuation on E tamely branching at p. Let w be the (unique p-henselian) prolongation of v to L. If w is still coarser than the canonical p-henselian valuation on L then, by Proposition 2.10, w | F = v | F is p-henselian with non-p-divisible value group and char F v ≠ p. If, on the other hand, w is strictly finer than the canonical p-henselian valuation w_{0} on L then Lw_{0} = Lw_{0}(p). As char Lw_{0} ≠ p and L ≠ L(p) this implies Γ_{w_{0}} ≠ pΓ_{w_{0}} and so w_{0} | F does the job.

2.7 Maximal abelian normal subgroups of G_{F} (p). As an immediate consequence of Theorem 2.15, we shall see that G_{F} (p) has maximal abelian normal subgroups. They are described in the following

Corollary 2.17 ([13], Cor. 2.3 and Cor. 3.3 and [15], section 4) Let F be a field with ζ_{p} ∈ F. Let v be the finest coarsening of the canonical p-henselian valuation on F with char F v ≠ p and let T_{v} denote the inertia subgroup of G_{F} (p) w.r.t. v. Then:

(a) There is a (unique) maximal normal abelian subgroup N ◁G_{F} (p) containing all normal abelian subgroups of G_{F} (p).

(b) G_{F} (p) is infinite abelian iff G_{F v}(p) = 1 or μ_{p^{∞}} ⊆ F v and G_{F v}(p) ≅ ℤ_{p}.

(c) If 1 ≠ N ≠ G_{F} (p), then N = T_{v} or p = 2, G_{F v}(2) ≅ ℤ_{2} ⋊ ℤ/2ℤ, μ_{2^{∞}} ⊆ F v(\sqrt{−1}) and N = G_{F (\sqrt{−1})}(2).

Proof If G_{F v}(p) ≅ 1 then G_{F} (p) ≅ T_{v} is abelian.

If G_{F v}(p) ≇ 1, ℤ_{p} or ℤ_{2} ⋊ℤ/2ℤ then G_{F v}(p) does not have any non-trivial normal abelian subgroup (otherwise, Theorem 2.15 would induce a proper p-henselian refinement of v with residual characteristic ≠ p) and so any normal abelian subgroup of G_{F} (p) becomes trivial under the canonical projection G_{F} (p) → G_{F v}(p), i.e., it is contained in the abelian kernel N = T_{v}. In particular, G_{F v}(p) is then non-abelian.

If G_{F v}(p) ≅ ℤ_{p} then G_{F} (p) ≅ T_{v} ⋊ ℤ_{p} is abelian iff μ_{p^{∞}} ⊆ F v iff μ_{p^{∞}} ⊆ some finite extension of F v iff there is some normal abelian subgroup N′ of G_{F} (p) not contained in T_{v} (note that T_{v} ∩ N′ is inertia subgroup of N′).

If G_{F v}(p) ≅ ℤ_{2} ⋊ℤ/2ℤ and T_{v} ≠ 1 then G_{F v(\sqrt{−1})}(2) ≅ ℤ_{2} and, as in the previous case, either μ_{2^{∞}} ⊆ F v(\sqrt{−1}) and G_{F (\sqrt{−1})}(2) is (maximal) abelian subgroup of G_{F} (2) or μ_{2^{∞}} ⊈ F v(\sqrt{−1}) and T_{v} is maximal abelian normal subgroup of G_{F} (2). Note that, by [3], Satz 2, G_{F v}(p) ≅ ℤ_{2} ⋊ ℤ/2ℤ is never abelian.

3 Henselianity inherited from Sylow fields

We have seen in Proposition 2.10 that henselianity, i.e., the property of a field to admit a non-trivial henselian valuation, which, of course, always goes up to algebraic extensions, sometimes also passes down from the top field to the bottom field of an algebraic extension, e.g., if the top field is not separably closed and if the extension is normal or finite. We will now see another perhaps surprising instance: If the fixed field of a non-trivial Sylow subgroup of G_{F} is henselian then so is F (with one exception).

Proposition 3.1 Let F be a field, let K be the fixed field of a non-trivial Sylow-p-subgroup P = G_{K} of G_{F} in F^{sep}.

(a) Let v be any coarsening of the canonical henselian valuation on K and, if p = 2 and Kv is real closed, assume v to be the coarsest henselian valuation with real closed residue field. Then the restriction v_{F} := v | F is a coarsening of the canonical henselian valuation on F. So, in particular, v_{F} is henselian.

(b) If K admits a non-trivial henselian valuation, then so does F, unless K is real closed and G_{F (\sqrt{−1})} is procyclic of odd order.

For the proof, we will need the following

Lemma 3.2 Let (F, v) be a valued field of mixed characteristic (0, p) with ζ_{p} ∈ F. Then for any a ∈ O_{v}:

1 + (1 − ζ_{p})^{p}a ∈ F^{p} ⇒ ∃x ∈ F v : x^{p} − x − \overline{a} = 0

The converse ‘⇐’ holds, when (F, v) is p-henselian.

Proof Obviously, 1 + (1 − ζ_{p})^{p}a ∈ F^{p} iff the polynomial

f(X) := (X + \frac{1}{1 − ζ_{p}})^{p} − ((\frac{1}{1 − ζ_{p}})^{p} + a)

has a zero in F. Now the lemma follows from the fact that f(X) ∈ O_{v}[X] induces the separable polynomial \overline{f}(X) = X^{p} − X − \overline{a} ∈ F v[X].

This is so because v(1 − ζ_{p}) = \frac{1}{p−1}v(p) and because \frac{p}{(1−ζ_{p})^{p−1}} ∈ −1 + M_{v}: Let g(X) be the irreducible polynomial of ζ_{p} over ℚ, i.e., g(X) = X^{p−1} + ⋯ + 1. Then

p = g(1) = (1 − ζ_{p})(1 − ζ_{p}^{2}) ⋯ (1 − ζ_{p}^{p−1})

= (1 − ζ_{p})^{p−1} · \underbrace{1 · (1 + ζ_{p}) ⋯ (1 + ζ_{p} + ⋯ + ζ_{p}^{p−2})}_{≡1·2···(p−1)≡(p−1)!≡−1 mod M_{v}} .

Proof of Proposition 3.1 (a) By Proposition 2.10, we may replace F by a finite subextension of K/F. Hence we may assume that ζ_{p} ∈ F if char K ≠ p, that F ≠ F (p) and, if Kv ≠ Kv^{sep}, that F v_{F} ≠ F v_{F} (p). Since v is a coarsening of the canonical henselian valuation on K, it suffices to show that v_{F} is henselian: it will then automatically be a coarsening of the canonical henselian valuation on F.

Case 1: Kv ≠ Kv^{sep} and Kv is not real closed Assume v_{F} is not henselian. Then there is a finite subextension L/F of K/F admitting distinct prolongations v_{1} = v | L and v_{2} of v_{F}. Note that Lv_{2} ≠ Lv_{2}(p) since F v_{F} ≠ F v_{F} (p) and since Kv is not real closed.

For any field E we write

E^{(p)} := \begin{cases}E^{p} & if char E ≠ p, \\ \{x^{p} − x | x ∈ E\} & if char E = p. \end{cases}

Choose a ∈ O_{v_{1}}, b ∈ O_{v_{2}} with a + M_{v_{1}} ∈ Lv_{1}^{(p)} and b + M_{v_{2}} ∉ Lv_{2}^{(p)}. By weak approximation, we then find an element x ∈ (a + M_{v_{1}}) ∩ (b + M_{v_{2}}). If char K = char Kv then x ∈ K^{(p)} \ L^{(p)} and if (char K, char Kv) = (0, p) then, by the previous lemma, 1 + (1 − ζ_{p})^{p} · x ∈ K^{p} \ L^{p}. So in any case, there is a subextension L′/L of K/L of degree p. But K is fixed field of a Sylow-p subgroup of G_{L}: contradiction.

Case 2: v is of (archimedean) rank 1 Assume again that v_{F} is not henselian and let L/F be a finite subextension of K/F with two distinct prolongations v_{1} = v | L and v_{2} of v_{F}. Let (L̂, v̂) be a henselisation of (L, v_{2}). Then L̂ ≠ L̂(p): note that if Kv is real closed then, by assumption, the non-trivial valuation v has no proper coarsening with real closed residue field, and so Γ_{v} ≠ 2Γ_{v}. Pick a ∈ L^{(p)} and b ∈ L̂ \ L̂^{(p)}. Since v_{1}, v_{2} are distinct rank-1 valuations we have strong approximation. So we find x ∈ L ∩ K^{(p)} \ L̂^{(p)} by choosing x close to a w.r.t. v_{1} and close to b w.r.t. v_{2} (L is dense in L̂ w.r.t. v_{2}, again because of ‘rank 1’). Hence, there is, once more, a subextension of K/L of degree p: contradiction.

Case 3: Kv = Kv^{sep} and Γ_{v} has a non-trivial convex archimedean subgroup Δ Let v′ be the coarsening of v with Γ_{v′} = Γ_{v}/Δ. Then v induces a rank-1 valuation v/v′ on Kv′ with Γ_{v/v′} = Δ. Since v is (a coarsening of) the canonical henselian valuation on K, Kv′ ≠ Kv^{′sep}, and since Kv = Kv^{sep}, Kv′ cannot be real closed either.

Thus, by case 1, v′_{F} := v′ | F is henselian. Since G_{Kv′} is a Sylow-p subgroup of G_{F v′_{F}}, Kv′ is the fixed field of a Sylow-p subgroup of G_{F v′_{F}} or (if p = char Kv′) the perfect hull thereof, and thus, by case 2, v_{F} /v′_{F} is also henselian (prolongations to the perfect hull are unique). Hence v_{F} is henselian.

Case 4: Kv = Kv^{sep} and Γ_{v} has no non-trivial convex archimedean subgroup Then O_{v_{F}} is the intersection of all valuation rings corresponding to proper coarsenings of v_{F}. Since they, again by case 1, are all henselian, so is v_{F}.

Case 5: Kv is real closed Then either K is real closed and v and v_{F} are trivial (hence henselian). Or K is not real closed and Γ_{v} has no non-trivial convex 2-divisible subgroup. But then the unique prolongation of v to K(\sqrt{−1}) is the canonical henselian valuation on K(\sqrt{−1}) restricting, by case 3 and 4, to a coarsening of the canonical henselian valuation on F (\sqrt{−1}). Hence, by Proposition 2.10 applied to F (\sqrt{−1})/F, v_{F} is henselian.

(b) If K admits a non-trivial henselian valuation then the canonical henselian valuation is non-trivial and any non-trivial coarsening v with Kv not real closed induces a non-trivial henselian valution on F.

If Kv is real closed and Γ_{v} ≠ 2Γ_{v} we may divide Γ_{v} by its maximal (2-)divisible convex subgroup to assume that no such non-trivial subgroup exists. Then v | F is henselian as in case 5.

If, finally, Kv is real closed and Γ_{v} = 2Γ_{v} then K is real closed and any formally real algebraic extension of F is Euclidean, i.e., has a unique ordering with the squares as positive elements. By [21], Lemma 5.1, this implies G_{F} ≅ A ⋊ ℤ/2ℤ, where A is abelian with 2 ∤ ♯A. If A is not procyclic then some Sylow-q subgroup of A is of the form ℤ_{q}^{α} with α > 1. Let K′ be the corresponding fixed field. Then, as char K = 0, ζ_{q} ∈ K′, and so, by Theorem 2.15, K′ admits a non-trivial henselian valuation v which is coarser than the canonical henselian valuation on K′ and with K′v not real closed. Since G_{K′} is a Sylow-q subgroup of G_{F}, v | F is then henselian.

4 Proofs of the main results

4.1 Galois characterization of henselian fields.

Proof of Theorem 1 For the non-trivial direction of Theorem 1 we assume that F is a field whose absolute Galois group G_{F} has, for some prime p, a Sylow-p-subgroup P with a non-trivial normal abelian subgroup N ◁ P. We further assume P to be not isomorphic to ℤ_{p}, ℤ/2ℤ or ℤ_{2}⋊ℤ/2ℤ. We will find a henselian valuation v on F tamely branching at p, i.e., with Γ_{v} ≠ pΓ_{v} and char F v ≠ p.

We first have to check that char F ≠ p. As in the proof of Theorem 2.15, P (and so G_{F}) has a subgroup of the shape ℤ_{p} ⋊ ℤ_{p}, say with fixed field L ⊆ F^{sep}, i.e., G_{L} = G_{L}(p) ≅ ℤ_{p} ⋊ ℤ_{p}. This cannot happen in characteristic p:

Lemma 4.1 Let L be a field with G_{L}(p) ≅ ℤ_{p} ⋊ ℤ_{p}. Then char L ≠ p.

Proof This is immediate from Galois cohomology: if char L = p then, by [42], II.2.2, Prop. 3, cd_{p}G_{L} ≤ 1, so

cd_{p}G_{L} ≤ 1 < 2 = cd_{p}ℤ_{p} ⋊ ℤ_{p}.

To keep the treatment elementary, however, we can also give a direct argument: Assume char L = p and G_{L}(p) ≅ ℤ_{p} ⋊ ℤ_{p}. Then, as in the proof of Lemma 2.16, any open subgroup of G_{L}(p) is of the same shape, (so, in particular, of rank 2) and the Kummer isomorphism of 𝔽_{p}-vector spaces appears in additive Artin-Schreier form:

L/L^{(p)} → Hom(G_{L}(p), 𝔽_{p}),

where L^{(p)} := {x^{p} − x | x ∈ L}. As all this holds for any finite subextension of L(p)/L, we may assume that 𝔽_{p} ⊆ L^{(p)}. The contradiction we are heading for is now obtained by the following fact: If a, b ∈ L are 𝔽_{p}-independent mod L^{(p)}, then so are α, β, γ ∈ L(α) mod L(α)^{(p)}, where α^{p} − α = a, β = b and γ = αβ.

To prove this fact, assume that for some i, j, k ∈ 𝔽_{p} and for some δ = c_{0} + c_{1}α + ⋯ + c_{p−1}α^{p−1} ∈ L(α) (c_{i} ∈ L),

iα + jβ + kγ = δ^{p} − δ.

We have to show that i = j = k = 0. Replace β, γ and δ to obtain

jb + (i + kb)α = c_{0}^{p} − c_{0} + c_{1}^{p}α^{p} − c_{1}α + ⋯ + c_{p−1}^{p}(α^{p−1})^{p} − c_{p−1}α^{p−1}
= c_{0}^{p} − c_{0} + c_{1}^{p}(a + α) − c_{1}α + ⋯ + c_{p−1}^{p}(a + α)^{p−1} − c_{p−1}α^{p−1}
= c_{0}^{p} − c_{0} + c_{1}^{p}a + c_{2}^{p}a^{2} + ⋯ + c_{p−1}^{p}a^{p−1}

+ (c_{1}^{p} − c_{1} + 2c_{2}^{p}a + 3c_{3}^{p}a^{2} + ⋯ + (p − 1)c_{p−1}^{p}a^{p−2})α

+ (c_{2}^{p} − c_{2} + \binom{3}{2}c_{3}^{p}a + ⋯ + \binom{p − 1}{2}c_{p−1}^{p}a^{p−3}) α^{2}

⋮

+ (c_{p−2}^{p} − c_{p−2} + \binom{p − 1}{p − 2}c_{p−1}^{p}a) α^{p−2}

+ (c_{p−1}^{p} − c_{p−1})α^{p−1}.

Now compare the coefficients of α^{p−1}, α^{p−2}, …:

α^{p−1} : c_{p−1}^{p} − c_{p−1} = 0, i.e., c_{p−1} ∈ 𝔽_{p}
α^{p−2} : c_{p−1} = 0 & c_{p−2} ∈ 𝔽_{p}, as a ∉ L^{(p)}
α^{p−3} : c_{p−1} = c_{p−2} = 0 & c_{p−3} ∈ 𝔽_{p}
⋮
α^{2} : c_{p−1} = c_{p−2} = ⋯ = c_{3} = 0 & c_{2} ∈ 𝔽_{p}
α^{1} : i + kb = c_{1}^{p} − c_{1} + 2c_{2}^{p}a
⇒ k = c_{2} = 0 & c_{1}^{p} − c_{1} = i, as 𝔽_{p} ⊆ L^{(p)}

Hence iα + jβ = δ^{p} − δ. Computing the trace tr of L(α)/L yields

tr(iα + jβ) = ia = (tr δ)^{p} − tr δ ∈ L^{(p)}.

So i = k = 0, and, thus, also j = 0 (by Artin-Schreier-Kummer theory, b ∉ L(α)^{(p)}).

To finish the proof of Theorem 1, denote the fixed field of P by K and observe that char K ≠ p implies ζ_{p} ∈ K. Hence, by Theorem 2.15, K admits a tamely branching henselian valuation v. We may also assume that v is a regular coarsening of the canonical henselian valuation on K. In particular, if p = 2 and Kv is real closed, no proper coarsening of v has real closed residue field. Hence, by Proposition 3.1, v | F is also henselian and, of course, still tamely branching.

As an immediate consequence of the proof of Theorem 1 and of Corollary 2.17 we obtain

Corollary 4.2 Let F be a field and let P be a pro-p (not necessarily Sylow) subgroup of G_{F}. Then:

(a) There is a unique maximal normal abelian subgroup N_{P} ◁ P containing all other normal abelian subgroups of P.

(b) Assume P ∈ P_{p}(G_{F}), i.e., N_{P} ≠ 1 and P ≇ ℤ_{p}, ℤ/2ℤ or ℤ_{2} ⋊ ℤ/2ℤ. Then there is a unique finest regular tamely branching henselian valuation v on the fixed field K of P. Moreover, if T_{v} denotes the inertia subgroup of P w.r.t. v, then:

[P : N_{P}] > 2  ⇔ v is anabelian (in this case N_{P} = T_{v});
[P : N_{P}] = 2  ⇔ either N_{P} = T_{v} and Kv is real closed
or G_{Kv} ≅ ℤ_{2} ⋊ ℤ/2ℤ and μ_{2^{∞}} ⊆ Kv(\sqrt{−1});
P = N_{P}  ⇔ either N_{P} = T_{v} and G_{Kv} ≅ 1
or G_{Kv} ≅ ℤ_{p} and μ_{p^{∞}} ⊆ Kv.

Proof If P ≅ 1, ℤ_{p}, ℤ/2ℤ or ℤ_{2} ⋊ℤ/2ℤ then, clearly, P has a maximal normal abelian subgroup. The same is true if P has no non-trivial abelian normal subgroup at all (then N_{P} = 1).

If P has a non-trivial normal abelian subgroup and P ≇ ℤ_{p} then, as in the proof of Theorem 1, char K ≠ p and so ζ_{p} ∈ K. Hence we may apply Corollary 2.17 to P = G_{K}(p).

Note that the finest tamely branching henselian valuation w on K and the finest regular tamely branching henselian valuation v on K are group-theoretically indistinguishable: v is a coarsening of w with Γ_{w/v} divisible, T_{v} = T_{w} and G_{Kv} ≅ G_{Kw}.

4.2 Group theoretic characterization of decomposition subgroups of G_{F}. Before we give a proof of Theorem 2, let us isolate three simple arguments which will be used throughout the proof:

Lemma 4.3 If (F, v) is a valued field then, for a fixed prime p, all Sylow-p-subgroups of decomposition subgroups of G_{F} w.r.t. v are conjugate in G_{F}.

Proof This is immediate from the fact that any two decompostion subgroups of G_{F} w.r.t. v are conjugate in G_{F} and from the fact that any two Sylow-p-subgroups of a profinite group D are conjugate in D.

Lemma 4.4 Assume that H ≤ H′ ≤ G are profinite groups with H ≠ H′. Then H contains no conjugate of H′ in G.

Proof There is some open normal subgroup N ◁ G with HN ≠ H′N. So \overline{G} := G/N is a finite group with subgroups \overline{H} := HN/N ≤ \overline{H}′ := H′N/N such that \overline{H} ≠ \overline{H}′. Now assume there is some x ∈ G with x^{−1}H′x ⊆ H. Let \overline{x} := xN ∈ \overline{G}. Then \overline{x}^{−1}\overline{H}′\overline{x} ⊆ \overline{H}. Hence ♯\overline{H} < ♯\overline{H}′ = ♯(\overline{x}^{−1}\overline{H}′\overline{x}) ≤ ♯\overline{H}: contradiction.

Lemma 4.5 Let (F, v) be a valued field with non-divisible Γ_{v} and assume that G_{F} ≅ ℤ_{p} or that G_{F} ≅ ℤ_{2} ⋊ ℤ/2ℤ and Γ_{v} has no non-trivial convex divisible subgroup. Then v is henselian.

Proof As Γ_{v} is non-divisible, the inertia subgroup of a decomposition subgroup of G_{F} w.r.t. v is infinite, hence, by the shape of G_{F}, of finite index in G_{F}. So a henselisation (L, w) of (F, v) is a finite extension of F. If w is a coarsening of the canonical henselian valuation u on L then, by Proposition 2.10, v is henselian. If w is strictly finer than u then Lw and Lu are separably closed and Γ_{w/u} is a nontrivial convex divisible subgroup of Γ_{w} ≅ Γ_{v}. Hence G_{F} ≅ ℤ_{p} and the restriction of u to F (again denoted by u) is then henselian with residue field F u separably closed. Thus v/u has to be henselian and so v is henselian as well.

Proof of Theorem 2(a) [and 2(b) ‘⇐’] We shall simultaneously prove Theorem 2(a) and the indicated direction of Theorem 2(b), putting the additional information required for the proof of 2(b) in square brackets […].

Let v be an [anabelian] regular tamely branching valuation, let S be a set of primes at which v tamely branches such that O_{v} = ⋂_{p∈S} O_{v_{p}}, where, for each p ∈ S, v_{p} is a non-trivial coarsening of v such that Γ_{v_{p}} has no non-trivial convex pdivisble subgroup [and such that G_{F v_{p}} has either non-procyclic Sylow-p-subgroups ≇ ℤ_{2} ⋊ ℤ/2ℤ or G_{F v_{p}} ≅ ℤ_{p} and μ_{p^{∞}} ⊈ F v_{p}(ζ_{p}) or G_{F v_{p}} ≅ ℤ_{2} ⋊ ℤ/2ℤ and μ_{2^{∞}} ⊈ F v_{2}(\sqrt{−1})]. Since those v_{p} are pairwise comparable we can choose one henselisation (F_{p}, w_{p}) of F in F^{sep} w.r.t. each v_{p} in such a way that all those F_{p} are pairwise comparable. Set D_{p} := G_{F_{p}} and D := ⋂_{p∈S} D_{p}. Then D is the decomposition subgroup of G_{F} w.r.t. v corresponding to the henselisation ⋃_{p∈s} F_{p} of (F, v).

Now fix p ∈ S and let P be a Sylow-p-subgroup of D_{p}. We have to show that D_{p} ∈ D(P) and that P ∈ P_{p,max} [and P ∈ P_{p,max}^{anab}].

If we assume that D_{p} ∉ D(P) then there is a proper subextension L/F of F_{p}/F such that P is a Sylow-p-subgroup of G_{L}. Since Γ_{w_{p}} (≅ Γ_{v_{p}}) has no non-trivial convex p-divisible subgroup, the unique prolongation u_{p} of w_{p} to the fixed field K of P is a coarsening of the canonical henselian valuation on K, and, if p = 2 and Ku_{p} is real closed, then u_{p} is the coarsest henselian valuation with this property. Hence, by Proposition 3.1, u_{p} | L = w_{p} | L is henselian, and hence, by Lemma 4.3, some conjugate of G_{L} in G_{F} is contained in D_{p} ⊂_{≠} G_{L}: but this contradicts Lemma 4.4.

Since v, and hence v_{p}, is tamely branching at p, we have P ∈ P_{p}. Now assume P ∉ P_{p,max}, and choose some P′ ∈ P_{p} with P′ ≻ P, i.e., P ⊂_{≠} P′ and N_{P} = N_{P′} ∩ P. Let K′ be the fixed field of P′, let u′ be the finest regular tamely branching henselian valuation on K′, let u be the unique prolongation of u′ to K, and let u′_{p} := u_{p} | K′. Then u and u_{p} are comparable. Moreover, u′_{p} is not henselian: otherwise, by Lemma 4.3, a conjugate of P′ in G_{F} would be contained in P, which is impossible, again by Lemma 4.4. So u is a proper coarsening of u_{p}.

Since u_{p} is regular, the inertia subgroup of P w.r.t. u is properly contained in the inertia subgroup of P w.r.t. u_{p} and, in particular, properly contained in N_{P} = N_{P′} ∩ P. This implies that N_{P′} is not the inertia subgroup of P′ w.r.t. u′. So, by Corollary 4.2, G_{K′u′} ≅ ℤ_{p} (and P′ is abelian) or G_{K′u′} ≅ ℤ_{2} ⋊ ℤ/2ℤ. As u_{p} | F = (u_{p} | K′) | F = u′_{p} | F = v_{p}, Γ_{u′_{p}} has no non-trivial convex p-divisible subgroup, and the same applies to the non-trivial valuation u′_{p}/u′ on K′u′. So, by Lemma 4.5, u′_{p}/u′ is henselian, and, thus, so is u′_{p}, which we have seen to be impossible. Hence P ∈ P_{p,max}.

[That P ∈ P_{p,max}^{anab} is clear since G_{F v_{p}} has either non-procyclic Sylow-p-subgroups ≇ ℤ_{2} ⋊ ℤ/2ℤ or they are ≅ ℤ_{p} and μ_{p^{∞}} ⊈ F v_{p}(ζ_{p}) or they are ≅ ℤ_{2} ⋊ ℤ/2ℤ and μ_{2^{∞}} ⊈ F v_{2}(\sqrt{−1}). Hence, by Corollary 4.2, [P : N_{p}] > 2.]

Proof of Theorem 2(b) ‘⇒’ Let D ≤ G_{F} be an anabelian abstract decomposition subgroup, so there is a non-empty set S of primes, and for each p ∈ S there is some P ∈ P_{p,max}^{anab} and some D_{p} ∈ D(P) such that the subgroups D_{p} of G_{F} (p ∈ S) form a linear chain w.r.t. inclusion and D = ⋂_{p∈S} D_{p}.

We first show that, for each p ∈ S, D_{p} is a decompostion subgroup of G_{F} w.r.t. some valuation v_{p} on F. To this end, fix p ∈ S, let F_{p} be the fixed field of D_{p} ∈ D(P) in F^{sep} and let K be the fixed field of P ∈ P_{p,max}^{anab}. Then, by Corollary 4.2, K admits a unique finest regular tamely branching henselian valuation u_{p} which is anabelian and N_{P} is the inertia subgroup of P w.r.t. u_{p}. Since Γ_{u_{p}} has no non-trivial p-divisible convex subgroup, we may apply Theorem 3.1 to deduce that w_{p} := u_{p} | F_{p} is henselian.

We claim that (F_{p}, w_{p}) is a henselisation of F w.r.t. v_{p} := w_{p} | F = u_{p} | F. Ohterwise there is a proper subextension F′_{p}/F of F_{p}/F such that w′_{p} := w_{p} | F′_{p} is henselian. Since D_{p} ∈ D(P) this implies that p | [F_{p} : F′_{p}], and so G_{F′_{p}} has a Sylow-p-subgroup P′ strictly containing P. Let K′ be the fixed field of P′, let u′_{p} := u_{p} | K′, and let T be the inertia subgroup of P′ w.r.t. u′_{p}. Since N_{P} is the inertia subgroup of P w.r.t. u_{p}, N_{P} = T ∩ P ⊆ N_{P′} ∩ P ⊆ N_{P} contradicting P ∈ P_{p,max}.

Γ_{u_{p}} has no non-trivial p-divisible convex subgroup. Therefore, the same applies to Γ_{v_{p}}.

For any p, q ∈ S with F_{q} ⊆ F_{p}, the unique (henselian) prolongation w of w_{q} to F_{p} is comparable to w_{p}, since w_{p}, by construction, is a coarsening of the canonical henselian valuation on F_{p}. Since {F_{p} | p ∈ S} is linearly ordered by inclusion, this implies that any two v_{p} (p ∈ S) are comparable. Hence O_{v} := ⋂ O_{v_{p}} is a valuation ring of F corresponding to a valuation v on F with henselisation ⋃_{p∈S} F_{p}, i.e., D is decomposition subgroup of G_{F} w.r.t. v.

4.3 Examples. Let us conclude with the three examples promised in the introduction, one showing that an abstract decomposition subgroup which is not anabelian may or may not be a (true) decomposition subgroup, one indicating that we cannot expect a group theoretic characterization of decomposition subgroups w.r.t. non-regular tamely branching valuations, and one example of an abelian anabelian (abstract=true) decomposition subgroup of an absolute Galois group.

Example 4.6 Let F = ℚ^{ab}((t)), where ℚ^{ab} denotes the maximal abelian extension of ℚ, and let v be the refinement of the power series valuation w on F by some p-adic valuation on ℚ^{ab}. Then F v ≅ 𝔽̃_{p}, Γ_{v} ≅ ℤ ⊕_{lex} \frac{1}{p^{∞}} ℤ and v is regular tamely branching (at all primes ≠ p). Let D be a decomposition subgroup of G_{F} w.r.t. v, so 1 ≠ D ≠ G_{F}. Then v is not encoded in G_{F}, i.e., there is a field F′ with G_{F′} ≅ G_{F} such that the only decomposition subgroups of G_{F′} w.r.t. any valuation on F′ are 1 and G_{F′}. In particular, the isomorphic copy D′ of D in G_{F′} is an abstract decomposition subgroup of G_{F′}, but not a decomposition subgroup w.r.t. any valuation on F′.

Proof G_{ℚ^{ab}} is projective. Hence, by [17], Theorem 20.15, there is a PACfield K which is a regular extension of ℚ^{ab} and such that res : G_{F} → G_{ℚ^{ab}} is an isomorphism. Let F′ = K((t)) and denote the power series valuation on F′ by w′. Then F ⊆ F′ and res : G_{F′} → G_{F} respects the ramification structure w.r.t. w and w′, i.e., there is a commutative diagram of canonical exact sequences

1 ⟶ T_{w′} ⟶ G_{F′} ⟶ G_{K} ⟶ 1
≅↓^{res}  ↓^{res}  ≅↓^{res}
1 ⟶ T_{w} ⟶ G_{F} ⟶ G_{ℚ^{ab}} ⟶ 1,

where T_{w} and T_{w′} (both ≅ ℤ̂) denote the inertia subgroups of G_{F} and G_{F′} w.r.t. w and w′.

Hence, res : G_{F′} → G_{F} is an isomorphism as well. Since w′ is a henselian rank-1 valuation, any non-trivial valuation on F′ with decomposition subgroup ≠ 1 is a refinement of w′. As PAC-field, K has only separably closed henselisations. Hence the only henselisations of F′ are F′ and F̃′.

Example 4.7 Let K = (ℚ^{ab} ∩ ℚ_{p})(p^{1/∞}) := ⋃_{n∈ℕ}(ℚ^{ab} ∩ ℚ_{p})( \sqrt[n]{p}), where ( \sqrt[n]{p})_{n∈ℕ} is a compatible sequence of roots of p, i.e., for m | n, ( \sqrt[n]{p})^{n/m} = \sqrt[m]{p}. Let v_{p} be the (unique prolongation of the) p-adic valuation on (ℚ^{ab} ∩ ℚ_{p} to) K. Then Γ_{v_{p}} ≅ ℚ and Kv_{p} ≅ 𝔽_{p}. Let F := K((t)) with power series valuation w and let v be the (non-henselian) refinement of w by v_{p}. Then Γ_{v} ≅ ℤ⊕_{lex} ℚ, v is tamely branching, no Sylow subgroup of any decomposition subgroup D of G_{F} w.r.t. v is abelian, but v is non-regular.

Then v is not encoded in G_{F}, i.e., there is a field F′ with G_{F′} ≅ G_{F} such that the only decomposition subgroups of G_{F′} are G_{F′} and 1.

Proof All we have to show is that G_{K} is projective. For then we can proceed as in the previous example. But projectivity of G_{K} is clear from the local-global-principle for the Brauer group of algebraic extensions of ℚ: any q-adic valuation on ℚ^{ab} ∩ ℚ_{p} for q ≠ p has residue field 𝔽̃_{q} (so projective decomposition subgroups), and all p-adic valuations on K have divisible value group (hence again projective decomposition subgroups).

Example 4.8 Let F = ℂ(t_{1}, t_{2}, …), where the t_{i} are indeterminates, and let v be the valuation with O_{v} = ⋂_{i∈ℕ} O_{v_{i}}, where v_{i} is defined recursively: v_{1} is the t_{1}-adic valuation on F with F v_{1} = ℂ(t_{2}, t_{3}, …), and if F v_{i} = ℂ(t_{i+1}, t_{i+2}, …) then v_{i+1} is the refinement of v_{i} by the t_{i+1}-adic valuation on F v_{i}, so F v_{i+1} = ℂ(t_{i+2}, t_{i+3}, …). Then v is anabelian regular tamely branching and the decomposition subgroup of G_{F} w.r.t. v is abelian.

Proof This is clear since F v = ℂ.

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