Finite by Presburger Groups and p-adic Fields

# Model theory of finite-by-Presburger Abelian groups and finite extensions of p-adic fields

Jamshid Derakhshan University of Oxford, Mathematical Institute, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK  and  Angus Macintyre Queen Mary, University of London, School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, UK
###### Abstract.

We define a class of pre-ordered abelian groups that we call finite-by-Presburger groups, and prove that their theory is model-complete. We show that certain quotients of the multiplicative group of a local field of characteristic zero are finite-by-Presburger and interpret the higher residue rings of the local field. We apply these results to give a new proof of the model completeness in the ring language of a local field of characteristic zero (a result that follows also from work of Prestel-Roquette).

###### Key words and phrases:
model theory, -adic numbers, local fields, model completeness, quantifier elimination, pre-ordered groups, Presburger arithmetic
###### 2000 Mathematics Subject Classification:
Primary 03C10, 03C60, 11D88, 11U09; Secondary 11U05

## 1. Introduction

A theory is called model-complete if for any model of and any , any definable subset of is defined by an existential formula. This concept was defined by Abraham Robinson (cf. [9]).

In this paper we define a class of pre-ordered abelian groups and prove that their theory is model-complete. Given a local field of characteristic zero , we show that certain quotients of the multiplicative group are finite-by-Presburger. We also show that they interpret the higher residue rings of the local field and other structure from the Basarab-Kuhlman language for valued fields. As an application of these results, we give a new proof of model completeness for a finite extension of a -adic field (a result that also follows from work of Prestel-Roquette) via result on first-order definitions of the valuation rings.

## 2. Finite-by-Presburger Abelian groups

We consider the language of group theory with primitives , together with a symbol standing for pre-order. The intended structures are abelian groups , 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 to mean and . This is obviously a congruence on , and the quotient is naturally an ordered abelian group. We restrict to the case when is a finite group . We call such finite-by-ordered. Note that the projection map

 G→G/∼

is pre-order preserving.

###### 1 Lemma.

is the torsion subgroup of if is finite-by-ordered.

###### Proof.

is torsion free.∎

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

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

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

###### 1 Theorem.

The theory of is determined by the theory of and the theory of the ordered group . Moreover, is decidable if and only if is decidable.

###### Proof.

Follows from the Feferman-Vaught Theorem [6].∎

We would like model-completeness of but settle here for a special case when is a model of Presburger arithmetic. Now Presburger arithmetic has quantifier elimination in the language with primitives , where denotes multiplication, is a constant interpreted as the minimal positive element, is an ordering, and is the subgroup of th 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 and , for all as above, and to the axioms of pre-ordered groups we add the following set of axioms for any given finite group . (In these axioms denotes the exponent of , and the torsion subgroup of .

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

ii) If and has order for some , then divides (we have a sentence for each ).

iii) , where denote a sentence that characterizes the group up to isomorphism (note that this sentence exists since is finite).

iv) If satisfies , then .

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

vi) The order on is trivial (i.e. for any two we have and ).

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

Clearly from above enriches to a model of these axioms.

###### 2 Theorem.

The theory determined by the above axioms is model-complete. It follows that is model-complete.

###### Proof.

Let be an embedding of models of the above axioms. We know as above that

 M2=H.Γ2

for some . Let . Then we have

 M1=H.Γ1.

Thus the embedding is the product embedding

 H.Γ1→H.Γ2.

Now is elementary (indeed, take in both copies of ), and

 Γ1→Γ2

is elementary since the map

 M1/H→M2/H

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

 H.Γ1→H.Γ2

is elementary. ∎

## 3. Groups of additive and multiplicative congruence classes

Let be a valued field. We shall denote by and the valuation ring and the valuation ideal respectively. We assume that has residue characteristic . We denote the value group of by . For an integer , set

 MK,k={a∈MK:v(a)>kv(p)},
 OK,k=OK/MK,k,

a local ring, and

 GK,k=K∗/1+MK,k,

a multiplicative group. denotes the canonical projection

 OK→OK,k,

and the canonical projection

 K∗→GK,k.

We denote by

 Θk⊆GK,k×OK,k

the binary relation defined by

 Θk(x,y)⇔∃z∈OK(π∗k(z)=x∧πk(z)=y).

We denote by the many-sorted structure

 (K,GK,k,OK,k,Θk).

Note that is well-defined on and surjective to the value group .

The groups are called the groups of multiplicative congruences and the rings are called the higher residue rings of . They occured in the work of Hasse on local fields. In model theory they first appeared in the language of Basarab [1] and then simplified by Kuhlmann [8]. His works with the many-sorted language

 (Lrings,Lgroups,Lrings,πk,π∗k,Θk),

for local fields. This has a sort for the field equipped with the language of rings, a sort for the groups equipped with the language of groups , and a sort for the residue rings equipped with the language of rings, for all . The language has symbols for the projection maps and and a predicate for the relation . We call this the language of Basarab-Kuhlmann and denote it by .

Note that does not have a symbol for the valuation on and on . However the valuation is quantifier-free definable from .

###### 2 Lemma.

Let be a finite extension of where is a prime. For any , the groups are pre-ordered -Presburger, where is the torsion group of .

###### Proof.

We first identify the torsion elements of . Clearly these must be of the form where . Note that

 gpf−1∈1+MK

and

 (gpf−1)pk∈1+MK,k.

Thus has (in ) order dividing , and if

 g∈1+MK,

then has order dividing in . Thus the torsion subgroup of has order . If denotes the group of units of . Then is the torsion subgroup of . Thus is isomorphic to which is the value group of , and hence is a -group, and so a model of Presburger arithmetic. ∎

###### 3 Theorem.

For any , the rings and the relation are interpretable in .

###### Proof.

Let denote an element of least positive value in (it follows that is also an element of least positive value in ). We let denote a generator of the cyclic group consisting of the Teichmuller representatives in (and hence the same holds for in ). has order . As before we have where and are respectively the residue field degree and ramification index of over .

An element of can be written uniquely in the form

 a+MK1,k,

where can be uniquely represented as

 ∑0≤j≤kcjπj

where are either or a power of . Similarly, an element of is uniquely of the form . Now except when all , these elements map to elements of (where ) under the map

 (∑0≤j≤kcjπj+MKi,k)→(∑0≤j≤kcjπj)(1+MKi,k).

This map is injective. Indeed, if two elements and map to the same element, then their difference lies in , but if and are different powers of , then by the usual Hensel Lemma argument that gives us the Teichmuller set, this gives a contradiction.

So we may construe the nonzero elements as constant elements of (and the same for ). We shall use the notation

 [∑0≤j≤kcjπj+MK1,k]

for them (similarly for ). We have a multiplication on these elements coming from the group , for , which we denote by . It is defined by

 [r1]⊙[r2]=[r1].[r2],

where is group multiplication in . We also have an addition on these elements together with the zero element coming from the ring , for , which we denote by . It is defined by

 [r1]⊕[r2]=[r1+r2].

We thus have a finite subset, denoted by (resp.  ), of (resp.  ) consisting of the nonzero elements

 [∑0≤j≤kcjπj+MK1,k]

(resp.  ) above together with the operations satisfying

 ([r1]⊕[r2])⊙[r3]=[r1]⊙[r1]⊕[r1]⊙[r3],

and the properties that is the unit element of and is the zero element.

Now, for , using Lemma 3, we can interpret in the relation as the set of all pairs satisfying the formula

 (r=[πk+1]∧v(g)≥k+1)∨
 ⋁s(0≤v(g)≤k∧v([s])=v(g)∧r=[s])),

where runs through the nonzero elements from before. (In fact, the satisfying the above is unique). Thus

 GKi,k×R

with the relation as above and with factors the two sorts is isomorphic to the structure

 GKi,k×OKi,k

with the relation and with factors the two sorts.∎

One has the following result of Basarab-Kuhlman on quantifier elimination.

###### 4 Theorem.

[8] Let be a Henselian valued field with characteristic zero and residue characteristic . Then given an -formula , there is an -formula which is quantifier free in the field sort such that for all

 K⊨φ(¯x)⇔Kk⊨ψ(¯x).

Note that for , is the residue field, and comes with an exact sequence

 1→k∗→GK,0→Γ→1.

We shall need a suitable description of the relation as follows.

###### 3 Lemma.

For any valued field and ,

 Θk={(g,α)∈GK,k×OK,k:(α=0∧v(g)≥k+1)∨(α≠0∧v(g)≤k)}.

Obvious. ∎

## 4. First-order definitions of valuation rings of local fields

We shall denote by the (first-order) language of rings with primitives . Given a structure , we let denote the -theory of , i.e., the set of all -sentences that are true in .

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

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

From the -definability of we easily get a -definition of the set

 {x:v(x)≤0},

and of the set of units . But it seems that no general nonsense argument gives a -definition of the maximal ideal .

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

 OK2∩K1=OK1,

whenever is an embedding of models of . We shall establish this condition for all embeddings of models of . For this, we shall first prove the following lemma.

###### 4 Lemma.

Let be an embedding of models of . Then

1. is relatively algebraically closed in ,

2. The valuation induced from on is Henselian.

###### Proof.

We first give a proof of (1). Suppose . Then , where is the ramification index and the residue field dimension (see [5],[2]). Clearly it is a first-order (but not yet visibly existential) property of (defined by ) expressed in the language of rings that the residue field has elements. Thus both and have residue fields (with respect to and ) of cardinality . (Recall, of course, that we do not yet know 4.0.1, so we have no natural map of residue fields). Similarly, in both and we have that is the th positive element of the value group (a condition that can be expressed by a first-order sentence using the formula defining the valuation).

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

So we must have , i.e.  is totally ramified over . Now we can assume that is a root of a monic Eisenstein (relative to ) polynomial over . Let

 F(x)=xe′+c1xe′−1+⋯+ce′.

Note that can not be Eisenstein over , for then it would be irreducible, and it has a root in .

Within the condition that is in the maximal ideal (for !) is simply that

 cejp−1∈OK1,

and the condition that is a uniformizing element is simply that both

 cee′p−1∈OK1,

and

 c−ee′p∈OK1,

hold. Now these conditions go up into since is a -formula. So

 cejp−1∈OK2

for all , and

 c−ee′p∈OK2.

Now (in the sense of ) is the th positive element of the value group (true in ). So in fact each (in the sense of ) for .

Since is not Eisenstein over , must fail to be a uniformizing element. But (in the sense of ), and is the th positive element of value group for , so does generate. So is relatively algebraically closed in . This proves (1).

We now prove (2). The valuation ring of the induced valuation on is , and its maximal ideal is . 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:=Xn+Xn−1+an−2Xn−2+⋯+a0

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

 MK2∩K1

of the induced valuation. Since all are in particular in , by Henselianity of and [5, Theorem 4.1.3, pp.88] we deduce that has a root in . Since by the first part, is relatively algebraically closed in , this must lie in , and by another application of [5, Theorem 4.1.3, pp.88] we deduce that is Henselian. The proof of the Lemma is complete. ∎

We can now prove the following.

###### 5 Lemma.

Let be an embedding of models of . Then

 (4.0.1) OK2∩K1=OK1.
###### Proof.

Consider the valuation ring in induced from . By Lemma 4, it is Henselian. Since any two Henselian valuation rings in are comparable, and has rank one value group (since its value group is a -group because it is elementarily equivalent to the value group of ), by [5, Theorem 4.4.1] the induced valuation on must agree with that given by and 4.0.1 follows.∎

It follows from Lemmas 1 and 2 that the valuation rings are -definable uniformly for models of .

### 4.1. Model completeness for a finite extension of Qp

In the case and , and in this case the multiplicative group of the residue field is isomorphic to the subgroup of th roots of unity in . If one has a cross-section , then is a subgroup of , and in any case (with cross-section or not) it is elementarily equivalent to . Note that the factor is definable as the set of -torsion elements.

So fix such an , with its attendant numbers with . For any field such that , the value group is a -group, and is the th positive element of the value group.

Now suppose is an extension of models of . Let be a uniformizing parameter for , i.e., is the least positive element if . By the preceding, is also a uniformizing element for .

###### 6 Lemma.

For any , where , the embedding of local rings

 OK1,k→OK2,k

is elementary.

###### Proof.

For any , where , the rings and have the same cardinality since and have the same finite residue field, so the inclusion is an isomorphism, and hence is elementary. ∎

###### 7 Lemma.

For any , where , the embedding of groups

 GK1,k→GK2,k

is elementary

###### 1 Remark.

In general, the theory of the structure is not model-complete.

Now we give a new proof of model completeness for a finite extension of . Let be a finite extension of . Let be an embedding of models of . We show that the embedding of in is elementary. Let be an -formula and consider where is a tuple from . By Theorem 4, there is a constant and an -formula which is quantifier-free in the field sort such that

 Th(L)⊢∀¯x(φ(¯x)↔ψ(¯x)).

Since and are models of , the formula holds in both and . Hence

 Ki⊨φ(¯a)↔ψ(¯a),

where . The subformula of from the field sort is quantifier free and so will hold in if and only if it holds in . Thus to prove that the inclusion of into is elementary, it suffices to consider the sub-formula of involving the sorts other than the field sort. In (for ), this formula is a Boolean combination of formulas of the sorts , formulas of the sorts , and formulas involving the relation for finitely many values of . We claim that each subformula of of each sort (including subformulas containing ) holds in if and only if it holds in . This would imply that holds in if and only if it holds in , which implies that holds in if and only if it holds in . To prove the claim, by Lemmas 6 and 7, the embedding of rings and the embedding of groups are both elementary for and any . Using the above interpretation of in (for ), we deduce that the embedding

 (K1,GK1,k,OK1,k,Θk)→(K2,GK2,k,OK2,k,Θk)

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

## References

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