Uniform definability of henselian valuation rings in the Macintyre language

Uniform definability of henselian valuation rings in the Macintyre language

Arno Fehm and Alexander Prestel

Abstract

We discuss definability of henselian valuation rings in the Macintyre languageLMac, the language of rings expanded bynth power predicates. In particular, we show that henselian valuation rings with finite or Hilbertian residue field are uniformly∃-∅-definable inLMac, and henselian valuation rings with value group Zare uniformly∃∀-∅-definable in the ring language, but not uniformly

∃-∅-definable inLMac. We apply these results to local fieldsQpandFp((t)), as well as to higher dimensional local fields.

1. Introduction

The question of definability of henselian valuation rings in their quotient fields goes back at least to Julia Robinson, who observed that the ring ofp-adic integersZp can be characterized inside the field ofp-adic numbersQp purely algebraically, for example, for odd prime numbers pas

Z_p={x∈Q_p: (∃y∈Q_p)(y²= 1 +px²)}.

This definition of the henselian valuation ring of the local fieldQpis existential (or diophantine) and parameter-free (∃-∅, for short), and it depends on p. For the local fields Fp((t)), an existential parameter-free definition of the henselian valuation ringFp[[t]] is much less obvious and was given only recently in [1]. Also this definition depends heavily onp.

Of particular importance in this subject and in applications to diophantine geometry and the model theory of fields is the question whether there areuniformdefinitions, for example, of Zp in Qp independent of p, and how complex such definitions have to be. It is known (see, for example, [3]) that there cannot be a uniform existential definition of Zp in Qp in the ring language Lring={+,−,·,0,1}, but partial uniformity results were obtained in [8].

Similarly, partially uniform existential definitions of valuation rings ofQplay a crucial role in the celebrated work [14].

Although the natural language to pose such questions is the ring language, in the study of the theory ofQ_p also the so-calledMacintyre language

LMac=Lring∪ {Pn:n∈N},

where eachP_n is a unary predicate symbol interpreted as the subset ofnth powers of the field, occurs naturally, cf. [17]. In this language, the following definition has recently been obtained in [3, Theorem 3] using results from the model theory of pseudo-finite fields.

Theorem 1.1 (Cluckers–Derakhshan–Leenknegt–Macintyre). There is an ∃-∅-formula in LMac that defines the valuation ring of every henselian valuation with residue field finite or pseudo-finite of characteristic not2.

Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-0-310541

We recall that a field is pseudo-finite if it is perfect, pseudo-algebraically closed (PAC) and has absolute Galois group ˆZ, the profinite completion ofZ. Since one can eliminate the predicates P_n by introducing new quantifiers, every LMac-definition gives rise to an Lring- definition. In particular, we have the following special case.

Corollary 1.2. There is an ∃∀-∅-formula in Lring that defines Zp in Qp and Fp[[t]] in Fp((t))for all odd prime numbersp.

The aim of this note is to discuss uniform definability of henselian valuation rings in the Macintyre language for families containing the local fieldsQp andFp((t)). Our results exploit both their specific (finite) residue fields and their (discrete) value groups.

A first generalization of Corollary 1.2 was already given by the second author in [15, Theorem 1]. Using an adaptation of the machinery developed there, we prove a definability result for p-henselian valuations in the Macintyre language (see Theorem 2.7), which in particular implies the following generalization of Theorem1.1.

Theorem 1.3. There is an ∃-∅-formula in the languageLMac that defines the valuation ring of every henselian valuation with residue field of characteristic not2 that is finite, PAC but not2-closed, or Hilbertian.

Note that ‘PAC but not 2-closed’ includes all pseudo-finite fields, but also fields like Q^tr(√

−1), where Q^tr is the field of totally real algebraic numbers. Note moreover that

‘Hilbertian’ includes in particular all global fields, like Q. The prime 2 is special here only in that Theorem2.7becomes particularly simple forp= 2, since the assumption that the field contains a primitivepth root of unity is always satisfied in this case.

In another direction, we generalize Corollary1.2by exploiting that the henselian valuations onQ_pandF_p((t)) have value groupZ. Here, an old result of Ax [2] shows that there is a uniform

∃∀∃∀-∅-definition inLringfor such valuations. We again work with p-henselian valuations and prove a result (Proposition3.6) that in particular improves Ax’ definition from∃∀∃∀to∃∀.

Theorem 1.4. There is an ∃∀-∅-formula in the languageLring that defines the valuation ring of every henselian valuation with value groupZ.

We also show that in this generality, the result cannot be improved further to give an existential definition in the Macintyre language (see Proposition4.6).

Theorem 1.5. The t-adic henselian valuation on C((t)) with value group Z cannot be defined by an∃-∅-formula in the languageLMac.

Finally, we also prove a variant (again forp-henselian valuations) that includes assumptions both on the residue field and on the value group (Theorem2.8). It implies, in particular, that thet-adic valuation on C((t)) can be defined by an∀-∅-formula in LMac, and it also implies the following theorem.

Theorem 1.6. There is an ∀-∅-formula in LMac that defines the valuation ring of every henselian valuation with value groupZand residue fieldFof characteristic not2with absolute Galois groupG_F ∼= ˆZ.

Combining this with Theorem1.1(or Theorem1.3), we summarize the following corollary.

Corollary 1.7. There are∃-∅and∀-∅-formulas inLMac that defineZ_p in Q_p andF_p[[t]]

inF_p((t))for every odd primep,although there are no such∃-∅or ∀-∅-formulas inLring. Since again we can eliminate the predicatesP_n, we observe that Corollary1.2holds with∃∀

replaced by∀∃, which can be deduced also from [15, Theorem 2].

Combining our positive and negative results, we acquire an almost complete understanding of theLMac-definability of henselian valuations on higher dimensional local fields in the sense of Parshin and Kato. We briefly discuss this in Section5.

2. Uniform definitions in the Macintyre language We will make use of the following general definability principle.

Proposition 2.1. Let L be a language containing Lring. Let Σ be a first-order axiom system inL ∪ {O},whereOis a unary predicate symbol. Then there exists anL-formulaϕ(x), defining uniformly in every model(K,O)ofΣthe setO, of quantifier type

∃if and only if(K1K2⇒ O1⊆ O2),

∀if and only if(K₁K₂⇒ O2∩K₁⊆ O1),

∃∀if and only if(K1≺∃K2⇒ O1⊆ O2),

∀∃if and only if(K1≺∃K2⇒ O2∩K1⊆ O1)

for all models(K1,O1),(K2,O2)ofΣ. Here, K1K2means thatK1 is anL-substructure of K2,andK1≺∃K2 means thatK1 is existentially closed inK2,asL-structures.

Proof. The detailed proof given in [15] for the special caseL=Lringgoes through verbatim for arbitraryL ⊇ Lring.

In particular, for the Macintyre language this implies the following corollary.

Corollary 2.2. LetΣbe a first-order theory of fields inLring∪ {O},whereOis a unary predicate symbol,and letN ⊆N. Then there exists an∃-∅-formula(respectively,∀-∅-formula) ϕ(x)inLring∪ {P_n:n∈N},defining uniformly in every model(K,O)ofΣthe setO,if and only ifO1⊆ O2(respectively,O2∩K₁⊆ O1)for all models(K₁,O1),(K₂,O2)ofΣfor which K₁is a subfield of K₂,and for alln∈N,(K₂^×)ⁿ∩K₁= (K₁^×)ⁿ.

Note that the condition (K₂^×)ⁿ∩K1= (K₁^×)ⁿis satisfied in particular whenK1is relatively algebraically closed inK2.

We fix some notation and recall a few definitions.

Definition 2.3. LetKbe a field andvbe a (Krull) valuation onK. We denote byO_vthe valuation ring ofv, bym_v its maximal ideal, by ¯K_v the residue field, and by Γ_v=v(K^×) the (additively written) value group ofv. The valuationv ishenselianif it has a unique extension to an algebraic closureK^alg of K, andp-henselian, forp a prime number, if it has a unique extension to the maximal Galois pro-pextensionK(p) ofK, cf. [7, Section 4.1]. We denote by ζ_p a primitive pth root of unity.

Lemma 2.4. Let(K1, v1)(K2, v2)be an extension of valued fields withchar(( ¯K1)_v1)=p, ζ_p∈K1 and(K₂^×)^p∩K1= (K₁^×)^p. Ifv2 isp-henselian,then so is v1.

Proof. Under the assumptions, v_i is p-henselian if and only if 1 +m_vi⊆(K_i^×)^p, cf. [7, Corollary 4.2.4]. So if v₂ is p-henselian, then 1 +m_v2 ⊆(K₂^×)^p, hence 1 +m_v1 = (1 +m_v2)∩ K₁⊆(K₂^×)^p∩K₁= (K₁^×)^p, which implies thatv₁ isp-henselian.

Trivially, every henselian valuation isp-henselian for everyp. The following two propositions generalize well-known results for henselian fields. Alternative proofs recently appeared in [12].

For the definition of a Hilbertian field, see [9, Chapter 12]; for the definition of a PAC field, see [9, Chapter 11].

Proposition 2.5. Ifvis a non-trivialp-henselian valuation on a fieldFwithchar( ¯F_v)=p andζ_p ∈F,thenF is not Hilbertian.

Proof. The proof of [9, Lemma 15.5.4] for henselian fields goes through in thep-henselian setting: Choosea∈F withv(a)>0 and letf(T, X) =X^p+aT−1,g(T, X) =X^p+T⁻¹−1.

If F is Hilbertian, then, since f and g are irreducible, there exists t∈F such that f(t, X), g(t, X) have no zero inF. However, both polynomials split overF(p), and at least one of them is inOv[X] and has a simple zero in the residue field, hence has a zero in F; see [7, Theorem 4.2.3].

Proposition 2.6. If F is PAC and v is a non-trivial p-henselian valuation on F, then F(p) =F.

Proof. Sincevisp-henselian, it has a unique extension toF(p), which we again denote byv.

SinceF is PAC it isv-dense inF(p); see [9, 11.5.3]. Thus, forσ∈Gal(F(p)|F) andx∈F(p), for everyγ∈Γ_v there existsa∈F withv(a−x)> γ, hencev(a−x^σ) =v^σ(a−x)> γ, since v^σ=v. Together, this implies thatv(x−x^σ)> γ, and as this holds for allγ, we conclude that x=x^σ. Thus,F =F(p).

Theorem 2.7. For every prime number p, there is an ∃-∅-formula in Lring∪ {P_p} that defines the valuation ring of everyp-henselian valued field(K, v)withζ_p∈Kand residue field F withchar(F)=pand

(a) F is finite,or

(b) F is PAC andF(p)=F, or (c) F is Hilbertian.

Proof. The valued fields as in the statement of the theorem form an elementary class axiomatized by some theory Σ: The class of p-henselian valued fields (K, v) withζ_p∈K and residue fieldF with char(F)=pcan be axiomatized, for example, using [7, Corollary 4.2.4].

Moreover, the class of finite or pseudo-finite fields (which is a subclass of (a) and (b)) is elementary, as are the fields in (b) and (c).

We want to apply Corollary2.2 to Σ. To this end, let (K1, v1) and (K2, v2) be such fields withK1 a subfield ofK2 and (K₂^×)^p∩K1= (K₁^×)^p. Denote bywthe restriction ofv2 to K1. By Lemma2.4,wisp-henselian.

The residue fieldF₁ of (K₁, v₁) satisfiesF₁(p)=F₁ in each of Cases (a)–(c): This is obvious in Case (a), holds by assumption in Case (b), and is well known in Case (c); see, for example, [9, 16.3.6]. Thus,v₁andware comparable by [13, Proposition 3.1].

Ifwisstrictly finerthanv₁, then it induces a non-trivialp-henselian valuation onF₁, which is a contradiction in each of Cases (a)–(c): In Case (a), because finite fields admit no non-trivial valuations at all, in Case (b) by Proposition2.6, and in Case (c) by Proposition2.5. Therefore, wiscoarserthanv₁, that is,O_v1 ⊆ O_w⊆ O_v2, as was to be shown.

Since every henselian valuation isp-henselian for everyp, Theorem1.3now follows from the special casep= 2.

Theorem 2.8. Letpbe a prime number andn∈Z₀. There is an ∀-∅-formula inLring∪ {P_p}that defines the valuation ring of everyp-henselian valued field(K, v)withζ_p∈K,residue fieldFthat satisfieschar(F)=pand|F^×/(F^×)^p|=pⁿ,and value group that does not contain ap-divisible convex subgroup.

Proof. Again, these valued fields form an elementary class axiomatized by some theory Σ, as above. We want to apply Corollary 2.2 to Σ. Let (K1, v1), (K2, v2) be models of Σ with K1⊆K2 and (K₂^×)^p∩K1= (K₁^×)^p, and denote by w the restriction of v2 to K1. By Lemma2.4,wisp-henselian. Denote byF1andF2the residue fields ofv1, respectively,v2. By assumption, dim_Fp(F₁^×/(F₁^×)^p) = dim_Fp(F₂^×/(F₂^×)^p) =n.

If v1 and w are incomparable, then they have a common coarsening with p-closed residue fieldF₀; see [13, Proposition 3.1]. The convex subgroup of Γ_v1 corresponding to the valuation induced byv₁onF₀is thenp-divisible (asF₀^×= (F₀^×)^p), contradicting the assumption.

Ifwis a proper coarsening ofv₁, then the valuation ¯v₁ induced byv₁on the residue fieldF ofwhas value group a convex subgroup of Γ_v1, hence notp-divisible. Therefore,

dim_Fp(F^×/(F^×)^p)dim_Fp(Γv¯1/pΓv¯1) + dim_Fp( ¯F_¯v^×₁/( ¯F_v¯^×₁)^p)>dim_Fp(F₁^×/(F₁^×)^p) =n.

Since (K₂^×)^p∩K₁= (K₁^×)^pandv₂isp-henselian, also (F₂^×)^p∩F= (F^×)^p: Indeed, ifx∈ O^×_w with ¯x= ¯y^p,y∈ O_v^×₂, then, sincef(T) =T^p−xsplits inK₂(p) and ¯f(T) has the simple zero y, there is¯ z∈K₂^× withz^p=x, sox∈(K₂^×)^p∩K₁= (K₁^×)^p, and thus ¯x∈(F^×)^p. Therefore, dim_Fp(F₂^×/(F₂^×)^p)dim_Fp(F^×/(F^×)^p)> n, contradicting the assumption. Thus, w is finer thanv₁, that is,O_v2∩K₁=O_w⊆ O_v1, as was to be shown.

For the t-adic valuation on K=C((t)), Theorem 2.8 immediately applies with n= 0 and arbitraryp. Moreover, Theorem1.6follows from the special casen= 1 andp= 2 of Theorem 2.8, sinceG_F ∼= ˆZimplies that|F^×/(F^×)²|= 2.

We note that while every∃-∅-definition of a valuation ring with finite residue fieldFq gives rise to an ∀-∅-definition of the same ring (see [1, Proposition 3.3]) it does not seem that this can be done in a uniform way, independent ofq.

3. Value groupZin the ring language

In this section, we will prove Theorem1.4in ap-henselian setting and for regular value groups.

Definition 3.1. An ordered abelian group Γ isdiscreteif it has a smallest positive element, p-regular if every quotient by a non-trivial convex subgroup isp-divisible, and regularif it is p-regular for every prime p. It is aZ-groupif it is discrete and regular.

An ordered abelian group Γ is a Z-group if and only if Γ≡Z as ordered groups; see [16, Theorem 4.1.3]. Examples ofZ-groups areZandZ⊕Q, where for ordered abelian groups Γ1, Γ2 we denote by Γ1⊕Γ2 the inverse lexicographic product.

For the rest of this section, we work in the following setting.

Setting 3.2. Let (K, v) be ap-henselian valued field and assume that one of the following cases holds:

(1) ζ_p∈K and char( ¯K_v)=p;

(2) char(K) =p;

(3) p= 2.

We also assume that the value group Γ = Γ_v is discrete and identify its smallest non-trivial convex subgroup withZ. Choose an elementt∈K withv(t) = 1∈Z⊆Γ.

In Case (1), letf(Y) =Y^p−1, in Cases (2) and (3) letf(Y) =Y^p−Y. Fora∈K,define R_a ={x∈K: (∃y∈K)(f(y) =ax^p)}.

Lemma 3.3. The setR_a contains allx∈K withpv(x)>−v(a).

Proof. If pv(x)>−v(a), then v(ax^p)>0, so the reduction off(Y)−ax^p has the simple zeroy= 1. In Case (1), the splitting field off(Y)−ax^pis a Kummer extension ofKcontained in K(p); in Case (2), the splitting field f(Y)−ax^p is an Artin–Schreier extension of K contained in K(p); in Case (3), the splitting field of f(Y)−ax^p is either K or a quadratic extension of K, hence contained in K(p). Thus, in each case, the fact that v is p-henselian implies that there existsy∈Kwithf(y) =ax^p, cf. [7, Theorem 4.2.3(2)].

Lemma 3.4. R_t=Ov.

Proof. By Lemma 3.3, O_v⊆R_t. If x, y∈K satisfy f(y) =tx^p, then x∈ O_v. Indeed, otherwise v(tx^p)<0. In Case (1), v(1 +tx^p) =v(tx^p)≡1 modpΓ, contradicting v(y^p)≡ 0 modpΓ. In Cases (2) and (3), v(tx^p)≡1 modpΓ, but v(y)<0, so v(f(y)) =v(y^p−y) = v(y^p)≡0 modpΓ, a contradiction.

For a subsetX ⊆K,let [X]ⁿ denote the set{x1· · ·x_n : x₁, . . . , x_n ∈X}. Define A={a∈K^× : 1∈R_a anda⁻¹∈/ [R_a]^p2}.

Forγ∈Γ,we letB_γ={x∈K:v(x)γ}. Thus,B0=Ov andB1=mv. Note thatB_δ·B_γ = B_δ+γ for allδ, γ∈Γ.

Lemma 3.5. Assume thatΓ is alsop-regular. Ifa∈A,thenR_a⊆ O_v.

Proof. Let a∈A. Note that 1∈R_a implies that R_a⊆[R_a]^p2. We do a case distinction according toγ=v(a)∈Γ:

γ <0 In this case,pv(a⁻¹) =−pγ >−γ, so Lemma3.3implies thata⁻¹∈R_a⊆[R_a]^p2, a contradiction.

γ= 0, . . . , p By Lemma 3.3, B₁⊆R_a. Suppose that R_a O_v, that is, there exists b∈R_a with v(b)−1. Then a⁻¹∈B_−p⊆B(p+1)v(b)+1=b^p+1·B₁⊆[R_a]^p+2⊆[R_a]^p2, a contradiction.

γ > p Since Γ isp-regular, there existk∈ {1, . . . , p}andα∈Γ such thatpα=γ−k. Then

−pα=−γ+k >−v(a), so, by Lemma3.3,B_−α⊆R_a. Thus,B_−p2α= [B_−α]^p2 ⊆[R_a]^p2. Note thatp²αγ: Ifγ=p+ 1, thenα= 1, so it holds; ifγp+ 2, thenpα=γ−kγ−pimplies that

p²αp(γ−p) =γ+ (p−1)γ−p²γ+ (p−1)(p+ 2)−p²=γ+p−2γ.

Thus,a⁻¹∈B_−p2α⊆[R_a]^p2, a contradiction.

Proposition 3.6. The∃3∀_2p²-∅-formulaϕ(x)in the languageLringgiven by (∃a, y, y0)(∀y1, . . . , y_p2, z1, . . . , z_p2) (¬(a= 0)∧f(y) =ax^p∧f(y0) =a∧

¬

⎛

⎝az₁· · ·z_p2 = 1∧

p²

i=1

f(y_i) =az^p_i

⎞

⎠

⎞

⎠

defines O_v in K for any p-henselian valued field (K, v) with discrete p-regular value group satisfying one of the three conditions(1)–(3).

Proof. Clearly, ϕ(K) =

a∈AR_a. By Lemma 3.5, this set is contained in Ov. Let t∈K with v(t) = 1. Then R_t=Ov (Lemma 3.4), so we have [R_t]^p2=Ov, and hence t⁻¹∈/[R_t]^p2. Thus,t∈A, henceϕ(K)⊇R_t=Ov, and therefore indeedϕ(K) =Ov.

Corollary 3.7. There is an∃∀-∅-formula inLringthat defines the valuation ring of every 2-henselian valuation with discrete2-regular value group.

Since every henselian valuation is in particular 2-henselian andZ is discrete 2-regular, this implies Theorem1.4.

Corollary 3.8. If (K, v) is a henselian valued field with value group Γ regular non- divisible,thenO_v is∃∀-∅-definable inLring.

Proof. The case where Γ is discrete follows from Corollary3.7. In the case where Γ is non- discrete, Hong [11, Theorem 4] gives a definition, which one can check to be∃∀: Indeed, the set Ψ defined there is ∃-{}-definable, thus so is Ω, hence mv=

=0Ω is ∀∃-∅-definable, which finally implies thatO_v= (K\m_v)⁻¹ is∃∀-∅-definable.

In fact, Hong does give a definition also in the case where Γ is discrete, but since in that case he builds on the argument of Ax, the definition he gets is at best∃∀∃. The assumption that Γ is non-divisible is, of course, necessary.

4. Value groupZin the Macintyre language

In this section, we prove our negative definability results, in particular Theorem1.5. Let (K, v) be a henselian valued field with value group Γ = Γ_v and residue fieldF = ¯K_vof characteristic zero. In order to prove thatOv is not∃-∅-definable inLMac, it suffices to construct henselian valued fields (K1, v1), (K2, v2) that are elementarily equivalent to (K, v) such that K1 is algebraically closed in K2 (since K1 is then an LMac-substructure of K2) and Ov1 Ov2. We first recall some standard definitions and facts.

Definition 4.1. For an ordered abelian group Γ, we denote by F((x^Γ)) the field of generalized power series _γ∈Γa_γx^γ with well-ordered support. The natural power series valuationv( _γ∈Γa_γx^γ) = min{γ:a_γ = 0}has value group Γ, residue fieldF and is henselian, cf. [6, Corollary 18.4.2]. As usual, we write F((x)) :=F((x^Z)) for the field of formal Laurent series. If Γ1,Γ2are ordered abelian groups, then there is a natural isomorphismF((x^Γ1⊕Γ2))∼= F((x^Γ₁¹))((x^Γ₂²)).

Construction 4.2. Let Δ be the divisible hull of Γ. We consider the power series fields K₁=F((x^Δ))((t^Γ))

with value groupu(K₁^×) = Δ⊕Γ and

K₂=F((s^Γ))((y^Δ))((z^Δ))

with value group v2(K₂^×) = Γ⊕Δ⊕Δ. Moreover, let F1:=F((x^Δ)) and denote by v1 the power series valuation on K1=F1((t^Γ)) with value group Γ and residue field F1. Define an embeddingφofK1 into the subfield

K0=F((s^Γ))((y^Δ))((z^Γ)) ofK₂as follows: For

f =

γ

f_γ(x)t^γ ∈K1

withf_γ(x)∈F1 for allγ let

φ(f) =

γ

f_γ(y)s^γz^γ ∈K₀.

This is indeed a homomorphism: For example, we can view it as the compositionφ=α◦of the canonical embedding:K1→K0 given by(x) =y, (t) =z, with the automorphism α ofK0that fixesF((s^Γ))((y^Δ)) and mapsα(z^γ) =s^γz^γ.

K₂=F

v2

((s^Γ))((y^Δ))((z^Δ))

K1=F

u ((x^Δ)) ((t^Γ)) v1

φ //K0=F((s^Γ))((y^Δ))((z^Γ))

F1=F((x^Δ)) F((s^Γ))((y^Δ))

F F((s^Γ))

F

Lemma 4.3. IfF((x^Q))≡F andΓ≡Γ⊕Q,then(K1, v1)≡(K2, v2)≡(K, v).

Proof. Note that Q≡Δ≡Δ⊕Δ since the theory of divisible ordered abelian groups is complete, cf. [16, Theorem 4.1.1]. Thus F((x^Q))≡F((x^Δ)) by the Ax–Kochen–Ershov theorem [16, Theorem 4.6.4], and Γ⊕Q≡Γ⊕Δ⊕Δ, since lexicographic products preserve elementary equivalence, cf. [10, Proof of 3.3]. Therefore,

( ¯K₁)_v1=F₁=F((x^Δ))≡F((x^Q))≡F= ( ¯K₂)_v2 =K_v and

Γ_v2= Γ⊕Δ⊕Δ≡Γ⊕Q≡Γ = Γ_v1= Γ_v.

Hence, since (K₁, v₁), (K₂, v₂),and (K, v) are henselian valued with residue field of character- istic zero, the Ax–Kochen–Ershov theorem implies that (K₁, v₁)≡(K₂, v₂)≡(K, v) as valued fields.

Lemma 4.4. φ⁻¹(Ov2) =Ou.

Proof. The definition ofφimplies thatφ(O_u)⊆ O_v2andφ(m_u)⊆m_v2: Indeed, for:K₁→ K₀this statement is obvious, andα:K₀→K₀leavesO_v2|K0 invariant. It follows thatφ(O_u) = Ov2∩φ(K1).

Lemma 4.5. The fieldφ(K₁)is algebraically closed inK₂.

Proof. By Lemma4.4, the embeddingφ:K1→K2 induces an embedding φ_∗: Γ_u= Δ⊕Γ−→Γ⊕Δ⊕Δ = Γ_v2

of value groups given by φ_∗(δ, γ) = (γ, δ, γ). Observe that φ_∗(Γ_u) is pure in Γ_v2: Indeed, if φ_∗(δ, γ) =n(γ1, δ1, δ2) withγ1∈Γ,δ1, δ2∈Δ, thenφ_∗(δ, γ) =nφ_∗(γ1, δ1)∈nφ_∗(Γ_u).

LetLbe a finite extension ofK₁ :=φ(K1) in K2. The pureness of the value groups implies that v2 is unramified in L|K₁, and both fields have the same residue field F. So since the henselian valued field (K₁, v2) of residue characteristic zero is algebraically maximal (see [7, Theorem 4.1.10]), we conclude thatL=K₁.

Proposition 4.6. If(K, v)is a henselian valued field with value groupΓ_v≡Γ_v⊕Qand residue field F of characteristic zero with F ≡F((Q)), then there is no∃-∅-formula in LMac

that defines the valuation ring ofv.

Proof. We apply the above construction and identify K₁ with φ(K₁)⊆K₂. ThenO_v2∩ K₁=O_u, and sinceO_uO_v1, this implies thatO_v1O_v2. Thus, (K₁, v₁) and (K₂, v₂) satisfy all properties listed at the beginning of this section, which concludes the proof.

Since Z≡Z⊕Q and C∼=C((Q)), Proposition 4.6 immediately applies to C((t)), thereby proving Theorem1.5. We will discuss more applications of Proposition4.6in the next section.

5. Higher dimensional local fields

In this last section, we briefly discuss the henselian valuations on higher dimensional local fields, by which we mean the following definition.

Definition 5.1. A (one-dimensional)local fieldis a completion of a number field (that is, a field isomorphic toR, C or a finite extension of Qp), or a completion of the function field of a curve over a finite field (that is, a field isomorphic to a finite extension of Fp((t))). An n-dimensional local field is a complete valued field with value group Z and residue field an (n−1)-dimensional local field.

Examples for two-dimensional local fields are R((t)), C((t)), Qp((t)) and Fp((t))((s)). An n-dimensional local fieldKcarries eitherk=nork=n−1 many different henselian valuations v1, . . . , v_k, where the value group ofv_k is a lexicographic product of k copies ofZ. (The fact that there are no other henselian valuations except for the obvious ones follows from F. K.

Schmidt’s theorem [7, Theorem 4.4.1].)

Lemma 5.2. If an ordered abelian groupΓhas a proper convex subgroupHsuch thatΓ/H is regular,thenΓ≡Γ⊕Q.

Proof. First of all, Γ≡H⊕Γ/H, cf. [10, Bottom of p. 282], and if Γ/H≡Γ/H⊕Q, then Γ⊕Q≡H⊕Γ/H⊕Q≡H⊕Γ/H≡Γ

since lexicographic products preserve elementary equivalence [10, Proof of 3.3]. Therefore, we can assume without loss of generality that Γ is regular. Since regularity is preserved under elementary equivalence (as follows, for example, from [5, Proposition 4]), some elementary extension Γ≺Γ^∗ has a proper convex subgroup H with Γ^∗/H divisible. (Alternatively, one could prove the regular case using the classical results of [18].) Thus, by the same reasoning as before, it suffices to prove the claim for Γ divisible. For Γ divisible, also Γ⊕Qis divisible, hence Γ≡Γ⊕Qsince the theory of divisible ordered abelian groups is complete [16, Theorem 4.1.1].

Example 5.3. Since all archimedean groups are regular, Lemma5.2implies that all ordered abelian groups Γ of finite rank satisfy Γ≡Γ⊕Q. This includes in particular the groups Z⊕ · · · ⊕Zthat occur as value groups of higher dimensional local fields.

Example 5.4. The condition F ≡F((Q)) is satisfied for the following fieldsF: (a) F is algebraically closed;

(b) F is real closed;

(c) F isp-adically closed;

(d) F admits a henselian valuationvwith Γ_v=Zand char( ¯F_v) = 0.

Indeed, in (a), F((Q)) is again algebraically closed and the theory of algebraically closed fields of fixed characteristic is complete. Similarly for (b) and (c). In (d), applying the Ax–

Kochen–Ershov theorem three times gives that F ≡F¯_v((Z))≡F¯_v((Z))((Q))≡F((Q)), since Z≡Z⊕Q.

It should now be clear that we get a complete understanding of the LMac-definability of the henselian valuations on all fields of the form F((t1)). . .((t_n)) where F is a local field of characteristic zero. Since the uniformity in Theorem2.8depends on|F^×/(F^×)²|, which is 1 for F =C, 2 forF =Rand 4 forF =Qp, we do not formulate a general result but rather discuss one family of examples in detail.

Example 5.5. The three-dimensional local fieldK=Q((t))((s)) has three non-trivial hen- selian valuations: The valuationv₁with value groupZand residue fieldQ((t)), the valuationv₂ with value groupZ⊕Zand residue fieldQ, and the valuationv₃ with value groupZ⊕Z⊕Z and residue fieldF. The definability of these valuations is as follows:

∃ inLMac ∀in LMac ∃∀in Lring ∀∃in Lring

v₁ No (a) Yes (d) Yes (g) Yes (i)

v₂ No (b) Yes (e) ? Yes (i)

v₃ Yes (c) Yes (f) Yes (h) Yes (i)

Here,Yesmeans ‘uniform for all odd prime numbers’, andNomeans ‘not even for a fixed’.

The question mark indicates that neither do we know thatv₂ is∃∀-definable in Lring for any fixed, nor do we know that there is no such definition that works uniformly for all.

Proof. (a) The value group of v₁ isZ≡Z⊕Q, and the residue field of v₁ isF =Q((t)), which carries a henselian valuation with value group Z and residue fieldQ of characteristic zero, henceF ≡F((Q)) by Example5.4(d). Therefore, Proposition 4.6applies.

(b) The value group of v₂ is Γ_v2=Z⊕Z, so Γ_v2≡Γ_v2⊕Q by Example5.3. The residue field ofv₂isF =Q, so F ≡F((Q)) by Example 5.4(c). So, again Proposition4.6applies.

(c) Sincev₃has finite residue fieldF, this follows from Theorem1.3.

(d) Since v₁ has residue field Q((t)) and |Q((t))^×/(Q((t))^×)²|= 8 by Hensel’s lemma, and Γ_v1 is discrete, we can apply Theorem2.8withp= 2 and n= 4.

(e) Sincev₂has residue fieldQand|Q^×/(Q^×)²|= 4 by Hensel’s lemma, and Γ_v2is discrete, we can apply Theorem2.8withp= 2 and n= 2.

(f) Sincev3has residue fieldFand|F^×/(F^×)²|= 2, and Γ_v3is discrete (so in particular has no non-trivial 2-divisible convex subgroup) we can apply Theorem2.8withp= 2 and n= 1.

(g) Sincev1has value groupZ, this is Theorem1.4.

(h) This follows from the fact that there is an∃-definition inLMac. (i) This follows from the fact that there is an∀-definition inLMac.

Acknowledgements. The authors would like to thank Will Anscombe and Franziska Jahnke for helpful comments on a previous version, and Immanuel Halupczok for some help with ordered abelian groups and the paper [4].

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