MODEL COMPLETENESS FOR FINITE EXTENSIONS OF p-ADIC FIELDS

JAMSHID DERAKHSHAN AND ANGUS MACINTYRE

Abstract. We prove that the first-order theory of a finite extension of the field of p-adic numbers is model-complete in the language of rings, for any prime p.

1. Introduction

This paper proves model-completeness in the language of ring theory for each finite extension of the p-adic field ℚ_{p}. Recall that a theory T is called modelcomplete if for any model M of T and any n ≥ 1, any definable subset of M^{n} is defined by an existential formula. This concept was defined by Abraham Robinson (cf. [11]).

1. Theorem. Let L be a finite extension of the field of p-adic numbers ℚ_{p}, where p is a prime. Then the first-order theory of L in the language of rings is model-complete.

Though we have a nagging feeling that we are neglecting something in the literature, we have not found any reference for such a model-completeness result. For ℚ_{p}, model-completeness in the language of rings is a well-known consequence of Macintyre’s quantifier elimination for p-adic fields in the Macintyre language (which is an extension of the language of rings by predicates for sets of nth powers, for all n) [9].

Quantifier elimination for a finite extension of ℚ_{p} was obtained by Prestel-Roquette [10] in an extension of the Macintyre language got by adding constant symbols for certain distinguished elements. However, the use of these constants does not readily give model-completeness.

Using relative quantifier elimination results of Basarab [1] and Kuhlmann [8] for the case of a fixed finite extension of ℚ_{p} in a many-sorted language involving sorts for higher residue ring and higher residue groups, we are reduced to proving model-completeness for these higher sorts. A key step, the model-completeness of the higher residue group sorts, is proved via defining a class of pre-ordered Abelian groups that we call finite-by-Presburger, and proving their model-completeness. We shall also interpret in such groups, each of the higher residue rings.

Given a valued field K, we shall denote the valuation by v, the valuation ring by O_{K}, and the maximal ideal by M_{K}. In [3], it was proved that given a finite extension K of ℚ_{p}, there is an existential definition of O_{K} without parameters in the ring language. In view of other results in [3] it is natural to ask if there is a universal definition of O_{K} without parameters in the ring language. Note that if we knew that the theory K is model complete, this and the existential definition of O_{K} would be immediate. In [3], a uniform parameter-free definition of O_{K} from the language of rings is given uniformly for all finite extensions of ℚ_{p} and all 𝔽_{q}((t)) for all primes p and prime powers q (and more generally uniformly for all fields with a valuation of arbitrary rank with finite or pseudo-finite residue field). This uniform definition is not existential (however it is ∃∀), and it is proved in [3] that there can not be any parameter-free uniform existential or universal definition from the language of rings. Interestingly, in our proof of Theorem 1, we first have to get the universal definition of valuation rings suggested above.

2000 Mathematics Subject Classification. Primary 03C10, 03C60, 11D88, 11U09; Secondary 11U05.

Key words and phrases. model theory, p-adic numbers, local fields, model completeness, quantifier elimination.

2. Proof of the Theorem

2.1. First-order definitions of valuation rings. We shall denote by L_{rings} the (first-order) language of rings with primitives {+, ., 0, 1}. Given a structure K, we let T h(K) denote the L_{rings}-theory of K, i.e., the set of all L_{rings}-sentences that are true in K.

Let L be a finite extension of ℚ_{p}, where p is a prime. By a theorem of F.K. Schmidt (cf. [5, Theorem 4.4.1]), any two Henselian valuation rings of L are comparable, so since L has a rank 1 valuation, it has a unique valuation ring O_{L} giving a Henselian valuation. By [3, Theorem 6], this valuation ring is defined by an existential L_{rings}- formula ψ(x). We remark that ψ(x) depends on the field L. For any field K which is elementarily equivalent to L, ψ(x) defines a valuation ring in K and hence a valuation.

By Krasner’s Lemma (see [2, Section 1]), L = ℚ_{p}(δ) for some δ algebraic over ℚ, and L has only finitely many extensions of each finite dimension. This property (with the same numbers) is true for any K which satisfies K ≡ L.

From the Σ_{1}-definability of O_{L} we easily get a Σ_{1}-definition of the set

{x : v(x) ≤ 0},

and of the set of units {x : v(x) = 0}. But it seems that no general nonsense argument gives a Σ_{1}-definition of the maximal ideal {x : v(x) > 0}.

We shall be working throughout in the language of rings, and our structures and morphisms and formulas are from this language unless otherwise stated.

Note that it is a necessary condition for model-completeness that

O_{K_{2}} ∩ K_{1} = O_{K_{1}},

whenever K_{1} → K_{2} is an embedding of models of T h(L). We shall establish this condition for all embeddings of models of T h(L). For this, we shall first prove the following lemma.

1. Lemma. Let K_{1} → K_{2} be an embedding of models of T h(L). Then (1) K_{1} is relatively algebraically closed in K_{2},

(2) The valuation induced from O_{K_{2}} on K_{1} is Henselian.

Proof. We first give a proof of (1). Suppose n = [L : ℚ_{p}]. Then n = ef, where e is the ramification index and f the residue field dimension (see [5],[2]). Clearly it is a firstorder (but not yet visibly existential) property of O_{L} (defined by ψ(x)) expressed in the language of rings that the residue field has p^{f} elements. Thus both K_{1} and K_{2} have residue fields (with respect to O_{K_{1}} and O_{K_{2}}) of cardinality p^{f}. (Recall, of course, that we do not yet know 2.1.1, so we have no natural map of residue fields). Similarly, in both K_{1} and K_{2} we have that v(p) is the eth positive element of the value group (a condition that can be expressed by a first-order sentence using the formula ψ(x) defining the valuation).

We now argue by contradiction. Suppose K_{1} is not relatively algebraically closed in K_{2}, then K_{1}(β) ⊂ K_{2}, for some β which is algebraic over K_{1} of degree m > 1. The valuation v of K_{1} defined by ψ(x) has a unique extension w to K_{1}(β) by Henselianity and [5, Theorem 4.4.1]. We have that m = e′f′, where e′ is the ramification index and f′ is the residue field dimension of K_{1}(β) over K_{1} with respect to w. (L satisfies all such equalities and so K_{1} does too. All this is of course with respect to the topology defined by ψ(x)). Now if f′ > 1 we may replace K_{1}(β) by its maximal subfield unramified over K_{1}. So we can in that case assume K_{1}(β) is unramified over K_{1}. Now K_{1} has residue field 𝔽_{p^{f}}, and then by Hensel’s Lemma K_{1}(β) contains a primitive (p^{ff′} − 1)th root of unity (similar arguments are used in [3]). So K_{2} contains a primitive (p^{ff′} − 1)th root of unity. But K_{2} certainly does not, since it’s residue field (with respect to ψ(x)) is 𝔽_{p^{f}} also.

So we must have f′ = 1, i.e. K_{1}(β) is totally ramified over K_{1}. Now we can assume that β is a root of a monic Eisenstein (relative to O_{K_{1}}) polynomial F (x) over K_{1}. Let

F (x) = x^{e′} + c_{1}x^{e′−1} + ⋯ + c_{e′}.

Note that F (x) can not be Eisenstein over K_{2}, for then it would be irreducible, and it has a root β in K_{2}.

Within K_{1} the condition that c_{j} is in the maximal ideal (for O_{K_{1}}!) is simply that

c_{j}^{e}p^{−1} ∈ O_{K_{1}},

and the condition that c_{e′} is a uniformizing element is simply that both

c_{e′}^{e}p^{−1} ∈ O_{K_{1}},

and

c_{e′}^{−e}p ∈ O_{K_{1}},

hold. Now these conditions go up into K_{2} since ψ(x) is a Σ_{1}-formula. So

c_{j}^{e}p^{−1} ∈ O_{K_{2}}

for all 1 ≤ j ≤ e′, and

c_{e′}^{−e}p ∈ O_{K_{2}}.

Now v(p) (in the sense of O_{K_{2}}) is the eth positive element of the value group (true in L). So in fact each v(c_{j}) > 0 (in the sense of O_{K_{2}}) for 1 ≤ j ≤ e′.

Since F (x) is not Eisenstein over K_{2}, c_{e′} must fail to be a uniformizing element. But ev(c_{e′}) = v(p) (in the sense of O_{K_{2}}), and v(p) is the eth positive element of value group for O_{K_{2}}, so c_{e′} does generate. So K_{1} is relatively algebraically closed in K_{2}. This proves (1).

We now prove (2). The valuation ring of the induced valuation on K_{1} is K_{1}∩O_{K_{2}}, and its maximal ideal is M_{K_{2}} ∩ K_{1}. By [5, Theorem 4.1.3, pp.88], Henselianity of a valued field is equivalent to the condition that any polynomial of the form

f := X^{n} + X^{n−1} + a_{n−2}X^{n−2} + ⋯ + a_{0}

where all the coefficients a_{j} are in the maximal ideal has a root in the field. So fix a polynomial f as above with the condition that the coefficients a_{j} are in the maximal ideal

M_{K_{2}} ∩ K_{1}

of the induced valuation. Since all a_{j} are in particular in M_{K_{2}}, by Henselianity of K_{2} and [5, Theorem 4.1.3, pp.88] we deduce that f has a root α in K_{2}. Since by the first part, K_{1} is relatively algebraically closed in K_{2}, this α must lie in K_{1}, and by another application of [5, Theorem 4.1.3, pp.88] we deduce that K_{1} is Henselian. The proof of the Lemma is complete. □

We can now prove the following.

2. Lemma. Let K_{1} → K_{2} be an embedding of models of T h(L). Then

(2.1.1)  O_{K_{2}} ∩ K_{1} = O_{K_{1}}.

Proof. Consider the valuation ring in K_{1} induced from O_{K_{2}}. By Lemma 1, it is Henselian. Since any two Henselian valuation rings in K_{1} are comparable, and K_{1} has rank one value group (since its value group is a ℤ-group because it is elementarily equivalent to the value group of L), by [5, Theorem 4.4.1] the induced valuation on K_{1} must agree with that given by O_{K_{1}} and 2.1.1 follows. □

It follows from Lemmas 1 and 2 that the valuation rings are ∀_{1}-definable uniformly for models of T h(L).

2.2. Basarab-Kuhlmann quantifier-elimination. We shall use the results of Basarab [1] and Kuhlmann [8] on relative quantifier elimination for a given finite extension of ℚ_{p}. It is a little bit easier to use Kuhlmann’s version [8]. Basarab works with categories of valued fields. Because of the previous results on definability of valuation rings we may work with models of T h(L) as above, imposing the valuation ring which is defined both existentially and universally, thereby giving automatically a category of valued fields. In Kuhlmann’s language [8] there is no symbol for valuation but the valuation can be defined in the language.

Recall the ingredients of the main result in [8]. In fact only the following special case of the formalism and result is needed. We shall use some convenient notation from [1].

Let K be a valued field with O_{K}, M_{K} as before. We suppose that K has residue characteristic p > 0. We denote the value group of K by Γ. For an integer k ≥ 0, set

M_{K,k} = {a ∈ M_{K} : v(a) > kv(p)},
O_{K,k} = O_{K}/M_{K,k},

a local ring, and

G_{K,k} = K^{∗}/1 + M_{K,k},

a multiplicative group. π_{k} denotes the canonical projection

O_{K} → O_{K,k},

and π_{k}^{∗} the canonical projection

K^{∗} → G_{K,k}.

Furthermore, we denote by

Θ_{k} ⊆ G_{K,k} × O_{K,k}

the binary relation defined by

Θ_{k}(x, y) ⇔ ∃z ∈ O_{K}(π_{k}^{∗}(z) = x ∧ π_{k}(z) = y).

We denote by K_{k} the many-sorted structure

(K, G_{K,k}, O_{K,k}, Θ_{k}).

Note that v is well-defined on G_{K,k} and surjective to the value group Γ. Kuhlmann’s language [8] for this structure is the many-sorted language

(L_{rings}, L_{groups}, L_{rings}, π_{k}, π_{k}^{∗}, Θ_{k}),

which has a sort for the field K equipped with the language of rings, a sort for the groups G_{K,k} equipped with the language of groups L_{groups}, and a sort for the residue rings O_{K,k} equipped with the language of rings, for all k ≥ 0. The language has symbols for the projection maps π_{k} and π_{k}^{∗} and a predicate for the relation Θ_{k}. We call this the language of Basarab-Kuhlmann and denote it by L_{BK}.

Note that L_{BK} does not have a symbol for the valuation on K and on G_{K,k}. However the valuation is quantifier-free definable from Θ_{k}.

2. Theorem. [8] Let K be a Henselian valued field with characteristic zero and residue characteristic p > 0. Then given an L_{BK}-formula φ(x̄), there is an L_{BK}- formula ψ(x̄) which is quantifier free in the field sort such that for all x̄

K ⊧ φ(x̄) ⇔ K_{k} ⊧ ψ(x̄).

Note that for k = 0, O_{K,k} is the residue field, and G_{K,k} comes with an exact sequence

1 → k^{∗} → G_{K,0} → Γ → 1.

We shall need a suitable description of the relation Θ_{k} as follows.

3. Lemma. For any valued field K and k ≥ 0,

Θ_{k} = {(g, α) ∈ G_{K,k} × O_{K,k} : (α = 0 ∧ v(g) ≥ k + 1) ∨ (α ≠ 0 ∧ v(g) ≤ k)}.

Proof. Obvious. □

In the case that interests us, K ≡ L and [L : ℚ_{p}] < ∞, and in this case the multiplicative group of the residue field is isomorphic to the subgroup μ_{p^{f} −1} of (p^{f} − 1)th roots of unity in K^{∗}. If one has a cross-section Γ → K^{∗}, then G_{K,0} is a subgroup of K^{∗}, and in any case (with cross-section or not) it is elementarily equivalent to μ_{p^{f} −1} ×Γ. Note that the μ_{p^{f} −1} factor is definable as the set of (p^{f} −1)- torsion elements.

So fix such an L, with its attendant numbers n, e, f with n = ef. For any field L such that K ≡ L, the value group is a ℤ-group, and v(p) is the eth positive element of the value group.

Now suppose K_{1} → K_{2} is an extension of models of T h(L). Let γ be a uniformizing parameter for K_{1}, i.e., v(γ) is the least positive element if v(K_{1}). By the preceding, γ is also a uniformizing element for v(K_{2}).

4. Lemma. For any k = mv(p), where m ≥ 0, the embedding of local rings

O_{K_{1},k} → O_{K_{2},k}

is elementary.

Proof. For any k = mv(p), where m ≥ 0, the rings O_{K_{1},k} and O_{K,k_{2}} have the same cardinality since K_{1} and K_{2} have the same finite residue field, so the inclusion O_{K_{1},k} → O_{K_{2},k} is an isomorphism, and hence is elementary. □

The proof of the next lemma is harder.

5. Lemma. For any k = mv(p), where m ≥ 0, the embedding of groups

G_{K_{1},k} → G_{K_{2},k}

is elementary

The proof Lemma 5 will be given in the next section. We shall deduce it from a more general theorem on model-completeness of a class of pre-ordered Abelian groups that we call finite-by-Presburger.

2.3. Model Theory of finite-by-Presburger Abelian groups. We consider the language of group theory with primitives {., 1,^{−1} }, together with a symbol ≤ standing for pre-order. The intended structures are abelian groups G, equipped with a binary relation ≤ satisfying

∀g (g ≤ g),
∀g∀h∀j (g ≤ h ∧ h ≤ j ⇒ g ≤ j),
∀g∀h (g ≤ h ∨ h ≤ g),
∀g∀h∀j (g ≤ h ⇒ gj ≤ hj).

It would be natural to call such structures pre-ordered abelian groups.

Define g ∼ h to mean g ≤ h and h ≤ g. This is obviously a congruence on G, and the quotient G/ ∼ is naturally an ordered abelian group. We restrict to the case when {g : g ∼ 1} is a finite group H. We call such G finite-by-ordered. Note that the projection map

G → G/ ∼

is pre-order preserving.

6. Lemma. H is the torsion subgroup of G if G is finite-by-ordered.

Proof. G/ ∼ is torsion free. □

Note that H is pure in G, indeed, if g ∈ G satisfies g^{m} ∈ H for some m, then g ∈ H. By [7, Theorem 7,pp.18], a pure subgroup of bounded exponent in an abelian group is a direct summand. Clearly H is of bounded exponent (being finite!), so H is a direct factor of G, so G = H.Γ, an internal direct product of subgroups, for some Γ.

Now Γ contains at most one element from each ∼-class, and the relation ≤ on Γ gives Γ the structure of an ordered abelian group. So in fact since G is the product of two pre-ordered groups, one of which H has only one ∼-class. So Γ ≅ G/H as ordered abelian groups.

Since G is a direct product of two pre-ordered groups, we have the following.

3. Theorem. The theory of (G, ≤) is determined by the theory of H and the theory of the ordered group (G/H, ≤). Moreover, G is decidable if and only if (G/H, ≤) is decidable.

Proof. Follows from the Feferman-Vaught Theorem [6]. □

We would like model-completeness of (G, ≤) but settle here for a special case when G/H is a model of Presburger arithmetic. Now Presburger arithmetic has quantifier elimination in the language with primitives {., 1,^{−1} , τ, P_{n}, ≤}, where . denotes multiplication, τ is a constant interpreted as the minimal positive element, ≤ is an ordering, and P_{n} is the subgroup of nth powers. Note that this is the multiplicative version of the usual formalism of Presburger arithmetic (cf. [4, Section 3.2, pp.197]).

So we augment the basic formalism of pre-ordered abelian groups with symbols τ and P_{n}, for all n ≥ 2 as above, and to the axioms of pre-ordered groups we add the following set of axioms for any given finite group H. (In these axioms m denotes the exponent of H, and T or(G) the torsion subgroup of G.

i) If the relation ≤ is an order, then τ is the minimal positive element, and if not, then τ = 1.

ii) If g ∈ G and g has order k for some k ∈ ℕ, then k divides m (we have a sentence for each k ≥ 1).

iii) T or(G) ⊧ σ, where σ denote a sentence that characterizes the group H up to isomorphism (note that this sentence exists since H is finite).

iv) If g ∈ G satisfies g ∼ 1, then g ∈ H.

v) G/T is totally ordered and is a model of Presburger arithmetic with τH the minimal positive element.

vi) The order ≤ on H is trivial (i.e. for any two g, h ∈ H we have g ≤ h and h ≤ g).

Note that given a model M of these axioms, H is the isomorphic to the torsion subgroup of M (by (iii)). Thus, given any finite group H, we obtain a theory which we denote by T_{H}. Note that if H = 1 (the identity group!), then T_{H} is the theory of Presburger arithmetic. We call these the axioms of pre-ordered groups with torsion H and ordered Presburger quotient modulo H.

Clearly G from above enriches to a model of these axioms.

4. Theorem. The theory determined by the above axioms is model-complete. It follows that (G, ≤) is model-complete.

Proof. Let M_{1} → M_{2} be an embedding of models of the above axioms. We know as above that

M_{2} = H.Γ_{2}

for some Γ_{2}. Let Γ_{1} := Γ_{2} ∩ M_{1}. Then we have

M_{1} = H.Γ_{1}.

Thus the embedding M_{1} → M_{2} is the product embedding

H.Γ_{1} → H.Γ_{2}.

Now H → H is elementary (indeed, take γ = 1 in both copies of H), and

Γ_{1} → Γ_{2}

is elementary since the map

M_{1}/H → M_{2}/H

is elementary because both ordered groups have the same minimal positive element. Therefore by the Feferman-Vaught Theorem [6] the map

H.Γ_{1} → H.Γ_{2}

is elementary. □

2.4. Proof of Lemma 5. Let K be a valued field and k ≥ 0. We first identify the torsion elements of G_{K,k}. Clearly these must be of the form g(1 + M_{K,k}) where v(g) = 0. Note that

g^{p^{f} −1} ∈ 1 + M_{K}

and

(g^{p^{f} −1})^{p^{k}} ∈ 1 + M_{K,k}.

Thus g has (in G_{K,k}) order dividing (p^{f} − 1)p^{k}, and if

g ∈ 1 + M_{K},

then g has order dividing p^{k} in G_{K,k}. Thus the torsion subgroup of G_{K,k} has order

(p^{f} − 1)(p^{f})^{me} = (p^{f} − 1)p^{nm}.

Now let the embedding K_{1} → K_{2} and k be as in Lemma 5. We have an inclusion

G_{K_{1},k} → G_{K_{2},k},

and the groups G_{K_{1},k} and G_{K_{2},k} have isomorphic torsion subgroups, say H. The group G_{K_{1},k}/H (resp. G_{K_{2},k}/H) is isomorphic to the value group of K_{1} (resp. value group of K_{2}). Hence G_{K_{1},k} and G_{K_{2},k} are ordered groups satisfying the axioms in Subsection 2.3 on pre-ordered groups with torsion H and ordered quotient modulo H. Furthermore, the minimal positive elements of G_{K_{1},k} and G_{K_{2},k} are the same. Thus by Theorem 4 the embedding G_{K_{1},k} → G_{K_{2},k} is elementary.

1. Remark. In general, the theory of the structure ℤ × (torsion subgroup) is not model-complete.

2.5. Completion of proof of Theorem 1. Let π denote an element of least positive value in K_{1} (it follows that π is also an element of least positive value in K_{2}). We let μ denote a generator of the cyclic group consisting of the Teichmuller representatives in K_{1} (and hence the same holds for μ in K_{2}). μ has order p^{f} − 1. As before we have k = ef where f and e are respectively the residue field degree and ramification index of L over ℚ_{p}.

An element of O_{K_{1},k} can be written uniquely in the form

a + M_{K_{1},k},

where a ∈ K can be uniquely represented as

∑_{0≤j≤k} c_{j}π^{j}

where c_{j} are either 0 or a power of μ. Similarly, an element of O_{K_{2},k} is uniquely of the form a + M_{K_{2},k}. Now except when all c_{j} = 0, these elements map to elements of G_{K_{i},k} (where i = 1, 2) under the map

( ∑_{0≤j≤k} c_{j}π^{j} + M_{K_{i},k}) → ( ∑_{0≤j≤k} c_{j}π^{j})(1 + M_{K_{i},k}).

This map is injective. Indeed, if two elements ∑_{0≤j≤k} c_{j}π^{j} and ∑_{0≤j≤k} c′_{j}π^{j} map to the same element, then their difference lies in M_{K_{i},k}, but if γ_{1} and γ_{2} are different powers of μ, then v(γ_{1} − γ_{2}) = 0 by the usual Hensel Lemma argument that gives us the Teichmuller set, this gives a contradiction.

So we may construe the nonzero elements ∑_{0≤j≤k} c_{j}π^{j} + M_{K_{1},k} as constant elements of G_{K_{1},k} (and the same for G_{K_{2},k}). We shall use the notation

[ ∑_{0≤j≤k} c_{j}π^{j} + M_{K_{1},k}]

for them (similarly for G_{K_{2},k}). We have a multiplication on these elements coming from the group G_{K_{i},k}, for i = 1, 2, which we denote by ⊙. It is defined by

[r_{1}] ⊙ [r_{2}] = [r_{1}].[r_{2}],

where . is group multiplication in G_{K_{i},k}. We also have an addition on these elements together with the zero element 0 coming from the ring O_{K_{i},k}, for i = 1, 2, which we denote by ⊕. It is defined by

[r_{1}] ⊕ [r_{2}] = [r_{1} + r_{2}].

We thus have a finite subset, denoted by R_{1} (resp. R_{2}), of G_{K_{1},k} (resp. G_{K_{2},k}) consisting of the nonzero elements

[ ∑_{0≤j≤k} c_{j}π^{j} + M_{K_{1},k}]

(resp. [∑_{0≤j≤k} c_{j}π^{j} + M_{K_{1},k}]) above together with the operations ⊕, ⊙ satisfying

([r_{1}] ⊕ [r_{2}]) ⊙ [r_{3}] = [r_{1}] ⊙ [r_{1}] ⊕ [r_{1}] ⊙ [r_{3}],

and the properties that [1] is the unit element of ⊙ and [π^{k+1}] is the zero element.

Now, for i = 1, 2, using Lemma 3, we can interpret in G_{K_{i},k} the relation Θ_{k} as the set Θ_{k}^{+} of all pairs (g, r) ∈ G_{K_{i},k} × R_{i} satisfying the formula

(r = [π^{k+1}] ∧ v(g) ≥ k + 1)∨

⋁_{s}(0 ≤ v(g) ≤ k ∧ v([s]) = v(g) ∧ r = [s])),

where s runs through the nonzero elements ∑_{0≤j≤k} c_{j}π^{j} + M_{K_{i},k} from before. (In fact, the s satisfying the above is unique). Thus

G_{K_{i},k} × R

with the relation Θ_{k}^{+} as above and with factors the two sorts is isomorphic to the structure

G_{K_{i},k} × O_{K_{i},k}

with the relation Θ_{k} and with factors the two sorts.

We can now finish the proof of model-completeness of T h(L). Let K_{1} → K_{2} be an embedding of models of T h(L). We show that the embedding of K_{1} in K_{2} is elementary. Let φ(x̄) be an L_{rings}-formula and consider φ(ā) where ā is a tuple from K_{1}. By Theorem 2, there is a constant N ≥ 0 and an L_{BK}-formula ψ(x̄) which is quantifier-free in the field sort such that

T h(L) ⊢ ∀x̄(φ(x̄) ↔ ψ(x̄)).

Since K_{1} and K_{2} are models of T h(L), the formula ∀x̄(φ(x̄) ↔ ψ(x̄)) holds in both K_{1} and K_{2}. Hence

K_{i} ⊧ φ(ā) ↔ ψ(ā),

where i = 1, 2. The subformula of ψ(ā) from the field sort is quantifier free and so will hold in K_{1} if and only if it holds in K_{2}. Thus to prove that the inclusion of K_{1} into K_{2} is elementary, it suffices to consider the sub-formula of ψ(ā) involving the sorts other than the field sort. In K_{i} (for i = 1, 2), this formula is a Boolean combination of formulas of the sorts O_{K_{i},k}, formulas of the sorts G_{K_{i},k}, and formulas involving the relation Θ_{k} for finitely many values of k. We claim that each subformula of ψ(ā) of each sort (including subformulas containing Θ_{k}) holds in K_{1} if and only if it holds in K_{2}. This would imply that ψ(ā) holds in K_{1} if and only if it holds in K_{2}, which implies that φ(ā) holds in K_{1} if and only if it holds in K_{2}. To prove the claim, by Lemmas 4 and 5, the embedding of rings O_{K_{1},k} → O_{K_{2},k} and the embedding of groups G_{K_{1},k} → G_{K_{2},k} are both elementary for k = m.v(p) and any m ≥ 0. Using the above interpretation of (G_{K_{i},k} × O_{K_{i},k}, Θ_{k}) in (G_{K_{i},k} × G_{K_{i},k}, Θ_{k}^{+}) (for i = 1, 2), we deduce that the embedding

(K_{1}, G_{K_{1},k}, O_{K_{1},k}, Θ_{k}) → (K_{2}, G_{K_{2},k}, O_{K_{2},k}, Θ_{k})

is elementary. This establishes the claim, and completes the proof.

References

1. Ş. Basarab, Relative elimination of quantifiers for Henselian valued fields, Ann. Pure Appl. Logic 53 (1991), no. 1, 51–74.

2. J. W. S. Cassels and A. Fröhlich (eds.), Algebraic number theory, London, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], 1986, Reprint of the 1967 original. MR 911121 (88h:11073)

3. Raf Cluckers, Jamshid Derakhshan, Eva Leenknegt, and Angus Macintyre, Uniformly defining valuation in henselian valued fields with finite or pseudofinite residue field, Annals of Pure and Applied Logic, to appear.

4. Herbert B. Enderton, A mathematical introduction to logic, second ed., Harcourt/Academic Press, Burlington, MA, 2001. MR 1801397 (2001h:03001)

5. A.J. Engler and A. Prestel, Valued fields, Springer Monographs in Mathematics, Springer-Verlag, 2005.

6. S. Feferman and R. L. Vaught, The first order properties of products of algebraic systems, Fund. Math. 47 (1959), 57–103. MR 0108455 (21 #7171)

7. Irving Kaplansky, Infinite abelian groups, University of Michigan Press, Ann Arbor, 1954. MR 0065561 (16,444g)

8. Franz-Viktor Kuhlmann, Quantifier elimination for Henselian fields relative to additive and multiplicative congruences, Israel J. Math. 85 (1994), no. 1-3, 277–306. MR 1264348 (95d:12012)

9. Angus Macintyre, On definable subsets of p-adic fields, J. Symbolic Logic 41 (1976), no. 3, 605–610. MR 0485335 (58 #5182)

10. A. Prestel and P. Roquette, Formally p-adic fields, Lecture Notes in Mathematics, vol. 1050, Springer-Verlag, 1984.

11. Abraham Robinson, Introduction to model theory and to the metamathematics of algebra, North-Holland Publishing Co., Amsterdam, 1963. MR 0153570 (27 #3533)

University of Oxford, Mathematical Institute, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK

E-mail address: derakhsh@maths.ox.ac.uk

Queen Mary, University of London, School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, UK

E-mail address: angus@eecs.qmul.ac.uk