RED FIELDS

A. BAUDISCH, A. MARTIN-PIZARRO, AND M. ZIEGLER

Abstract. We apply Hrushovski-Fraïssé’s amalgamation procedure to obtain a theory of fields of prime characteristic of Morley rank 2 equipped with a definable additive subgroup of rank 1.

1. Introduction

In the early 90’s E. Hrushovski came up with an interesting application of Fraïssé’s construction to obtain certain structures whose rank was described by a priori exhibited predimension function δ. Unfortunately, this procedure has remained an obscure area of model theory to the general audience, in spite of its relevant applications. Hrushovski’s amalgamation method is – in principle – a twostep process: first, one obtains a Fraïssé generic model from a certain class so that the above model exhibits some of the properties we are interested in, though of infinite rank. Secondly, one makes it collapse, i.e. we impose certain restrictions on the class we are amalgamating in, related to algebraizing certain types which have small δ-dimension but large rank in the aforementioned model. This relies on some combinatorial arguments in order to determine on advance the maximal number of realizations allowed for a given type within our class.

In [11] a theory of fields of rank ω · 2 with a definable additive subgroup of rank ω was constructed by applying the first part of the amalgamation method as aforementioned described. It was asked by by B. Poizat in [9] after Corollary 3.3 whether a similar structure of finite rank could exist. It is known that such a field cannot be of characteristic 0, because they have no definable non-trivial additive subgroups. In this paper, we will collapse the above structure and answer the previous question positively. Moreover, an explicit description of the axioms of the generic model obtained is given. The red additive subgroup has Morley rank 1 and the whole structure Morley rank 2.

The main source of this work is E. Hrushovski’s fusion of two strongly minimal sets [6]. Using Hrushovski’s paper B. Poizat [10] and J. Baldwin and K. Holland [1] produced a field with black points of rank 2. In [2] we gave an exposition of [6], simplifying some of the arguments, which allowed us in [3] to give a new construction of a field with black points of rank 2 with a simpler axiomatization. The methods used there will be developed further in this paper in order to obtain a field of rank 2 with a red additive subgroup. Similar ideas yield a fusion over a vector space over a finite field of two strongly minimal sets with DMP in [4].

Date: August 21, 2006.

1991 Mathematics Subject Classification. Primary: 03C65; Secondary:03C50.

Key words and phrases. Model Theory, Fields of finite Morley Rank.

blau.tex,v 2.7, August 21, 2006.

A. Martin-Pizarro conducted research supported by a Marie Curie research grant MEIF-CT-2005-009541 and a DFG-Forschungsstipendium MA3310/1-1.

The authors would like to thank the Newton Institute (Cambridge, UK) for the hospitality provided during the semester Model Theory and its applications to Algebra and Analysis, in which the three authors took part. Moreover, the second author would like to thank the Deutsche Forschungsgemeinschaft for its support during Spring Semester 2005.

2. Group sets

The following Lemma was proven in 1990 by the third author [12].

Lemma 2.1. Let G be a stable abelian group, and a, b and c in G be pairwise independent over B with a + b + c = 0. Then:

(1) Their strong types over B have the same stabilizer U. Furthermore, U is connected.

(2) a, b and c are generic elements of acl(B)–definable cosets of U. Moreover, if G is totally transcendental, it follows that a, b and c have the same Morley rank over B, namely MR(U), and U is definable over acl(B).

Note 2.2. Suppose that a ∈ U. Then −b and c have the same type over B, since they are generic in the same coset of U.

As in Section 2 in [3] we will encode types using codes. These codes will enable us to define the class of structures we are concerned with and later use them to give an axiomatization of the Fraïssé generic model.

We work inside 𝕌, a universal algebraically closed field of prime characteristic p and all formulae are L–formulas, where L denotes the language of rings. All sets in this section will be subsets of some cartesian power 𝕌^{n}, unless specified. We use deg_{M} to denote the Morley degree.

Definition 2.3. Let X be a definable subset of 𝕌^{n} with deg_{M}(X) = 1. We say that X is a group set (resp. torsor set) if its generic type is the generic type of a definable subgroup G (resp. coset of a subgroup) of (𝕌^{n}, +). If X is not a torsor set, then X is called groupless.

Notation 2.4. Two definable sets X and Y are equivalent, if MR(X) = MR(Y) and MR(X △ Y ) < MR(X). We write X ∼ Y.

Given A ⊂ 𝕌, we denote by ⟨A⟩ the subspace generated by A (considering 𝕌 as an 𝔽_{p}-vector space in the natural way).

Lemma 2.5 (Hasson, Hils [5]). With notation as above, the following holds:

(1) Given Y ∼ X, if X is a group set, so is Y.

(2) Given H in GL_{n}(𝔽_{p}), if X is a group set, so is H(X) = {Hx⃗ | x⃗ ∈ X}.

(3) The set { b⃗ | φ(𝕌, b⃗) is a group-set } is definable for every formula φ(x⃗, y⃗). Similarly for torsor sets.

Proof. The first two statements are clear.

For the third point, note that algebraically closed fields have the DMP. So deg_{M} φ(x⃗, b⃗) = 1 is an elementary property of b⃗. We claim that φ(𝕌, b⃗) is a group set iff “There exist two generic b⃗–independent realizations a⃗_{1} and a⃗_{2} of φ(x⃗, b⃗) whose sum a⃗_{1} + a⃗_{2} lies in φ(x⃗, b⃗).” This is easily seen to be an elementary property of b. One direction of the equivalence is clear. To prove the other direction let a⃗_{0}, a⃗_{1} have the property above and let k be the Morley rank of φ(x⃗, b⃗). We have MR(a⃗_{0} + a⃗_{1}/b⃗a⃗_{0}) = MR(a⃗_{1}/b⃗a⃗_{0}) = k and therefore MR(a⃗_{0} + a⃗_{1}/b⃗) ≥ k. ⊧ φ(a⃗_{0} + a⃗_{1}, b⃗) implies that MR(a⃗_{0} + a⃗_{1}/b⃗) = k, so a⃗_{0}, a⃗_{1}, a⃗_{0} + a⃗_{1} are pairwise independent over b⃗. And they all have the same type over b⃗. Hence by Lemma 2.1 tp(a⃗_{0}/b⃗) is a group type.

Note also that X is a torsor set if and only if for some x the set X − x is a group set. □

Definition 2.6. Given a group set X, its invariant group is the set Inv(X) = {H ∈ GL_{n}(𝔽_{p}) | H(X) ∼ X}.

Note that “H ∈ Inv(φ(x⃗, b⃗))” is an elementary property of b⃗.

Lemma 2.7. Let X be a B-definable set of Morley degree 1, and e⃗_{0} and e⃗_{1} two generic B-independent elements. If e⃗_{0} − He⃗_{1} | ⌣_{B} e⃗_{0} for some H in GL_{n}(𝔽_{p}), then

X is a torsor set. Moreover, if X is a group set, then H is in Inv(X).

Proof. From

MR(He⃗_{1}/B, e⃗_{0} − He⃗_{1}) = MR(e⃗_{0}/B, e⃗_{0} − He⃗_{1}) = MR(e⃗_{0}/B) ≥ MR(He⃗_{1}/B)

we obtain that {e⃗_{0}, He⃗_{1}, e⃗_{0} − He⃗_{1}} is a pairwise B-independent triple. By 2.1 X is an torsor set. If G is the group that corresponds to X, then e⃗_{0} and He⃗_{1} are generic elements of some cosets of G. That implies that H(G) = G. □

3. Prealgebraic Sets

Let us fix some notation. For any subset A of 𝕌 let ⟨A⟩ be the 𝔽_{p}–vector space generated by A, and dim(A) the dimension of ⟨A⟩. The relative dimension dim(A/B) is the dimension of ⟨A, B⟩/⟨B⟩. We write trdeg(A/B) for the transcendence degree of A over B.

Definition 3.1. Let n = 2k > 0 and X be a b⃗–definable subset of 𝕌^{n}, of Morley rank k and Morley degree 1. We call X prealgebraic (over the parameters b⃗) if for all generic a⃗ ∈ X, the following holds:

a) dim(a⃗) = n.

b) Given a non-trivial subspace U of ⟨a⃗⟩, then

(3.1)  2 · trdeg(U/b⃗) > dim(U).

Equivalently,

(3.2)  2 · trdeg(a⃗/Ub⃗)) < n − dim(U).

Lemma 3.2. If X is prealgebraic over b⃗ and a⃗ in X is generic, then dim(a⃗/b⃗) = n.

Proof. Clearly, ⟨a⃗⟩ ∩ acl(b⃗) = 0 by Equation (3.1). □

Lemma 3.3. Prealgebraicity is preserved under equivalence^{1}, translation and the action of GL_{n}(𝔽_{p}).

1in the sense of 2.4

From the first assertion it follows that prealgebraicity does not depend on the choice of parameters. We use this to prove the second statement.

Proof. Suppose X is prealgebraic defined over b⃗ and X ∼ Y with Y defined over b⃗′. Choose a generic element a⃗ of X ∩ Y. For a subspace U of ⟨a⃗⟩, we have that U | ⌣_{b⃗} b⃗′ and U | ⌣_{b⃗′} b⃗. So MR(U/b⃗) = MR(U/b⃗′) and Y is prealgebraic.

Let now m⃗ be in 𝕌^{n}. We want to show that X −m⃗ is again prealgebraic. Choose a⃗ in X, generic over m⃗, b⃗. Hence, a⃗ − m⃗ is generic in X − m⃗ over b⃗, m⃗. Given x in ⟨a⃗⟩ ∩ acl(b⃗m⃗), it follows from a⃗ | ⌣_{b⃗} m⃗ that x lies in acl(b⃗). By 3.2 x = 0. In

particular,

⟨a⃗⟩ ∩ acl(m⃗) = 0

Therefore, the coordinates of a⃗ − m⃗ are linearly independent. Let U ⊂ ⟨a⃗ − m⃗⟩ be a non-trivial subspace. There is some subspace V ⊂ 𝔽_{p}^{n} with

U = {v · (a⃗ − m⃗) | v ∈ V }

where · denotes the scalar product. By the above, dim U = dim V and we have 2 · trdeg(U/b, m) = 2 · trdeg(V a/b, m) = 2 · trdeg(V a/b) > dim(V a) = dim(U). The case of a transformation via an invertible matrix is clear. □

4. Codes

Definition 4.1. The formula φ(x⃗, y⃗) encodes X if there is some b⃗ in 𝕌 such that X ∼ φ(x⃗, b⃗). Definition 4.2. A code α is a tuple consisting of the following objects: natural numbers n_{α} and k_{α} and a formula φ_{α}(x⃗, y⃗) with the following properties:

(a) length(x⃗) = n_{α} = 2k_{α}.

(b) The set φ_{α}(x⃗, b⃗) is either empty or has Morley rank k_{α} and Morley degree 1.

(c) Let a⃗ be a realization of φ_{α}(x⃗, b⃗). Then dim(a⃗) = n_{α} and Equation (3.2) in Definition 3.1 holds for all non-trivial subspaces of ⟨a⟩.

(d) φ_{α}(x⃗, b⃗) ∼ φ_{α}(x⃗, b⃗′) ⟹ b = b′.

(e) If some non-empty φ_{α}(x⃗, b⃗) is groupless, then all φ_{α}(x⃗, b⃗′) are. (We call hence α a groupless or coset code accordingly.)

(f) φ_{α}(x⃗ + m⃗, b⃗) is encoded by φ_{α} for all m⃗.

(g) For all H in GL_{n_{α}}(𝔽_{p}), the set φ_{α}(Hx⃗, b⃗) is encoded by φ_{α}.

It follows from (d) that b⃗ is a canonical basis of the type of Morley rank k_{α} determined by φ_{α}(x⃗, b⃗). Moreover, either φ_{α}(x⃗, b⃗) is empty or prealgebraic by (c).

We first show that these codes enable us to encode prealgebraic sets. Lemma 4.3. Every prealgebraic set X ⊂ 𝕌^{n} is encoded by some code α as above. Proof. A straight-forward generalization of the argument exhibited in Lemma 2.3 [3] following [6] shows how to find such a formula φ_{0}(x⃗, b⃗) with properties (a)-(d) for a given set X as above.^{2}

By case 3 in Lemma 2.5, we can strengthen φ_{0} so that (e) holds. Let us call such a formula φ_{0}(x⃗, b⃗) a good formula.

2 For (c) we use that the dimension of ⟨a⟩ can be elementarily expressed, since 𝔽_{p} is finite.

Let T (x⃗) denote an affine transformation (compositum of an element of GL_{n}(𝔽_{p}) and a translation by some tuple m⃗). By Lemma 3.3 T (X) is again a set as in the statement, hence it can be encoded by a good formula. Therefore, by compactness there are finitely many φ_{1}, …, φ_{r} that encode all possible T (X)’s. Moreover, we may assume that either all (or none) encode groupless sets by 2.5. Choose now w_{1}, …, w_{r} different 𝔽_{p}-tuples with the same length. Define:

θ_{i}^{1}(b⃗) = “ No φ_{j} (j < i) encodes φ_{i}(x⃗, b⃗)”
θ_{i}^{2}(b⃗) = “ φ_{α_{i}}(x⃗, b⃗) is equivalent to some φ_{0}(Hx⃗ + m′, b⃗′) ”
φ′_{i}(x⃗, y⃗) = φ_{i}(x⃗, y⃗) ∧ θ_{i}^{1}(y⃗) ∧ θ_{i}^{2}(y⃗)

Finally, let

φ_{α}(x⃗, y⃗_{1}, y⃗) = ⋁_{i=1}^{r} (φ′_{i}(x⃗, y⃗) ∧ y⃗_{1} = w_{i})

It is clear that φ_{α} has properties (a)-(e). To show that (f) and (g) hold, let b⃗, m⃗ and H be given. By construction, φ_{α}(x⃗, b⃗) is equivalent to some φ_{0}(H′x⃗ + m′, b⃗′). Hence,

φ_{α}(Hx⃗ + m⃗, b⃗) ∼ φ_{0} ((H′H)x⃗ + (H′m⃗ + m′), b⃗′)

The above is again encoded by α by construction. □

Now, we can choose our set of representatives:

Theorem 4.4. There is a set C of codes such that any definable prealgebraic set X of Morley degree 1 is encoded by a unique α in C.

Proof. Let α_{i} be a list of all codes. Again, define:

θ_{i}(b⃗) = “ No α_{j} (j < i) encodes φ_{α_{i}}(x⃗, b⃗)”
φ_{α′_{i}}(x⃗, y⃗) = φ_{α}(x⃗, y⃗) ∧ θ_{i}(y⃗)

We need only show that the φ_{α′_{i}} still satisfy (f) and (g). By construction, φ_{α′_{i}}(Hx⃗ + m⃗, b⃗) is encoded by α_{i}. We need only show that no α_{j} with j < i encodes it. Suppose that

φ_{α_{i}}(Hx⃗ + m⃗, b⃗) ∼ φ_{α_{j}}(x⃗, b⃗′)

Then

φ_{α_{i}}(x⃗, b⃗) ∼ φ_{α_{j}}(H^{−1}x⃗ − H^{−1}m⃗, b⃗′) ∼ φ_{α_{j}}(x⃗, b⃗′′)

for some b⃗′′, which contradicts our definition of φ′_{α}. Finally set

C = { α′_{i} | i = 0, 1, … }.
□

For the rest of the paper we fix a set C of codes as given by the above theorem. We call the codes in C good codes.

We now choose for each α in C a natural number m_{α} such for each Morley sequence e⃗_{i} of φ_{α}(x⃗, b⃗) of length m_{α} and all b⃗′ with |b⃗′| = |b⃗| we have e⃗_{i} | ⌣_{b⃗} b⃗′ for

some i. We can always find m_{α} ≤ |b⃗| + 1.

Theorem 4.5. For each α ∈ C and λ ≥ m_{α} there is a formula Ψ_{α}(x⃗_{0}, …, x⃗_{λ}) with following properties:

(a) For any initial segment {e⃗_{0}, …, e⃗_{λ}, f} of a Morley sequence of φ_{α}(x⃗, b⃗),

Ψ_{α}(e⃗_{0} − f, …, e⃗_{λ} − f)

holds.

(b) For each realization (e⃗_{0}, …, e⃗_{λ}) of Ψ_{α} there is a unique b⃗ with ⊧ φ_{α}(e⃗_{i}, b⃗) for 0 ≤ i ≤ λ. Moreover, b⃗ ∈ dcl(e⃗_{i_{1}}, …, e⃗_{i_{mα}}) for any i_{1} < ⋯ < i_{m_{α}}. (We call b⃗ the canonical parameter of the sequence e⃗_{0}, …, e⃗_{λ}).

(c) Each realization of Ψ_{α} is 𝔽_{p}–linear independent.

(d) If ⊧ Ψ_{α}(e⃗_{0}, …, e⃗_{λ}), then for i ∈ {0, …, λ}:

⊧ Ψ_{α}(e⃗_{0} − e⃗_{i}, …, e⃗_{i−1} − e⃗_{i}, −e⃗_{i}, e⃗_{i+1} − e⃗_{i}, …, e⃗_{λ} − e⃗_{i})

(e) Given a realization (e⃗_{0}, …, e⃗_{λ}) of Ψ_{α} with canonical parameter b⃗ as in (b), we have the following:

Suppose α is groupless:

1) If e⃗_{i} generic in φ_{α}(x⃗, b⃗), then

e⃗_{i} − He⃗_{j} |̸ ⌣_{b⃗} e⃗_{i}

for all H in GL_{n_{α}}(𝔽_{p}) and j ≠ i.

Suppose α is an coset code, then:

2) φ_{α}(x⃗, b⃗) is a group-set.

3) Ψ_{α}(e_{0}, …, e_{i−1}, e_{i} − e_{j}, e_{i+1}, …, e_{λ}) for j ≠ i.^{3}

4) Ψ_{α}(e_{0}, …, e_{i−1}, He_{i}, e_{i+1}, …, e_{λ}) for all H in Inv(φ_{α}(x⃗, b⃗)).^{3}

5) Moreover, if e⃗_{i} is generic in φ_{α}(x⃗, b⃗), then

e⃗_{i} − He⃗_{j} |̸ ⌣_{b⃗} e⃗_{i}

for all j ≠ i and H in GL_{n_{α}}(𝔽_{p}) \ Inv(φ_{α}(x⃗, b⃗)).

In Section 7 we will assign a fixed large λ = μ(α) to every α.

Proof. Consider the following partial type

Σ(e⃗_{0}, …, e⃗_{λ}) = “ there is some b⃗′ and some Morley sequence a⃗_{0}, …, a⃗_{λ}, f⃗ of

φ_{α}(x⃗, b⃗′) with e⃗_{i} = a⃗_{i} − f⃗”.

Claim. Σ has properties (a)–(e).

Proof of the claim. By definition, Σ has property (a). Given a realization e⃗_{0}, …, e⃗_{λ} of Σ, there are some b⃗′ and a⃗_{0}, …, a⃗_{λ}, f⃗ as above. Hence, {e⃗_{i}}_{0≤i≤λ} is a Morley sequence of φ_{α}(x⃗ + f⃗, b⃗′). Then φ_{α}(x⃗ + f⃗, b⃗′) ∼ φ_{α}(x⃗, b⃗) for some b⃗ by (f) in Definition 4.2. Since b⃗ is the canonical base of the type determined by φ_{α}(x⃗, b⃗), the sequence {e⃗_{i}}_{0≤i≤λ} is a Morley sequence for φ_{α}(x⃗, b⃗). Given another b⃗^{∗} which satisfies φ_{α}(e⃗_{i}, y⃗) for m_{α} many i’s, there is some i such that

𝕌 ⊧ φ_{α}(e⃗_{i}, b⃗^{∗})

3By (b) and λ ≥ m_{α} this new sequence has also canonical parameter b⃗.

and

e⃗_{i} | ⌣_{b⃗} b⃗^{∗}

by the choice of m_{α}. It follows that φ_{α}(x⃗, b⃗^{∗}) ∼ φ_{α}(x⃗, b⃗) and by (d) in Definition 4.2, b⃗^{∗} = b⃗. Hence, (b) holds for Σ.

The linear independence in (c) follows from Lemma 3.2. We get

acl(b⃗, e⃗_{0}, …, e⃗_{i−1}) ∩ ⟨e⃗_{i}⟩ = 0.

Given a⃗_{0}, …, a⃗_{λ}, f⃗ as above, the sequence a⃗_{0} …, a⃗_{i−1}, f⃗, a⃗_{i+1}, …, a⃗_{λ}, a⃗_{i} is again Morley for φ_{α}(x⃗, b⃗′). Hence,

(a⃗_{0} − a⃗_{i}, …, a⃗_{i−1} − a⃗_{i}, f⃗ − a⃗_{i}, a⃗_{i+1} − a⃗_{i}, …, a⃗_{λ} − a⃗_{i}) ⊧ Σ

i.e.

(e⃗_{0} − e⃗_{i}, …, e⃗_{i−1} − e⃗_{i}, −e⃗_{i}, e⃗_{i+1} − e⃗_{i}, …, e⃗_{λ} − e⃗_{i}) ⊧ Σ

This yields (d).

For (e), if α is groupless, then φ_{α}(x⃗, b⃗) is not a torsor set and the result follows by Lemma 2.7. Otherwise, the set X = φ_{α}(𝕌, b⃗′) defines a torsor set. Hence, X − f⃗ ∼ φ_{α}(x⃗, b⃗) is a group set (note that f⃗ is in X). In the case of a group code, extend the Morley sequence {e⃗_{i}}_{0≤i≤λ} by an element d⃗. Therefore, the sequence e⃗_{0} + d⃗, …, e⃗_{i−1} + d⃗, e⃗_{i} − e⃗_{j} + d⃗, e⃗_{i+1} + d⃗, …, e⃗_{λ} + d⃗, d⃗ is again Morley for φ_{α}(x⃗, b⃗). Hence,

Σ(e⃗_{0}, …, e⃗_{i} − e⃗_{j}, …, e⃗_{λ}).

Likewise, given H in Inv(φ_{α}(x⃗, b⃗)), the sequence

e⃗_{0} + d⃗, …, e⃗_{i−1} + d⃗, He⃗_{i} + d⃗, e⃗_{i+1} + d⃗, …, e⃗_{λ} + d⃗, d⃗

is again Morley for φ_{α}(x⃗, b⃗). Therefore, Σ(e⃗_{0}, …, He⃗_{i}, …, e⃗_{λ}). The last point follows again from Lemma 2.7. □(Claim)

Take now some finite part Ψ′_{α} of Σ which implies (b), (c), (e1), (e2) and (e5) by compactness (note that (a) follows trivially).

If α is groupless, consider the following operations:

V_{i}(x⃗_{0}, …, x⃗_{λ}) = (x⃗_{0} − x⃗_{i}, …, x⃗_{i−1} − x⃗_{i}, −x⃗_{i}, x⃗_{i+1} − x⃗_{i}, …, x⃗_{λ} − x⃗_{i})

and let V be the subgroup generated by these operations (Observe that V has cardinality (λ + 1)!(λ + 2)).

Now, the formula

Ψ_{α}(x⃗_{0}, …, x⃗_{λ}) = ⋀_{V ∈V} Ψ′_{α}(V (x⃗_{0}, …, x⃗_{λ}))

satisfies (d), and lies also in Σ.

If α is an coset code, property (d) follows from (e3) and (e4). Hence, it is enough to find Ψ_{α} which satisfy the latter. Let W(x⃗_{0}, …, x⃗_{λ}) be the subgroup of GL_{n_{α}(λ+1)}(𝔽_{p}) generated by the operations mentioned in (e3) and (e4). Again, W is finite, and depends on Inv(φ_{α}(x⃗, b⃗)). Note that λ ≥ m_{α}, hence b⃗ remains constant after applying these operations by (b). Set therefore:

Ψ_{α}(x⃗_{0}, …, x⃗_{λ}) = ⋀_{W ∈W(x⃗_{0},...,x⃗_{λ})} Ψ′_{α}(V (x⃗_{0}, …, x⃗_{λ})),

which has the required properties. □

Definition 4.6. Let α, λ and Ψ_{α} be as above. A realization of Ψ_{α} is called a difference sequence for α.

Moreover, given a realization e⃗_{0}, …, e⃗_{λ} of Ψ_{α}, we denote by a derived difference sequence one obtained by composition of the following operations:

e⃗_{0} − e⃗_{i}, …, e⃗_{i−1} − e⃗_{i}, −e⃗_{i}, e⃗_{i+1} − e⃗_{i}, …, e⃗_{λ} − e⃗_{i}.

If ν ≤ λ and we use the above operation only for i ≤ ν then we speak about a ν-derived difference sequence .

Note 4.7. A general transposition (ij) (and therefore all permutations of {0, …, λ}) is obtained as follows:

e⃗_{0}, …, e⃗_{i}, …, e⃗_{j}, …, e⃗_{λ}
(V_{j})  e⃗_{0} − e⃗_{j}, …, e⃗_{i} − e⃗_{j}, …, −e⃗_{j}, …, e⃗_{λ} − e⃗_{j}
(V_{i}) e⃗_{0} − e⃗_{j} − (e⃗_{i} − e⃗_{j}), …, e⃗_{j} − e⃗_{i}, …, −e⃗_{j} − (e⃗_{i} − e⃗_{j}), …, e⃗_{λ} − (e⃗_{j} − (e⃗_{i} − e⃗_{j}) =

e⃗_{0} − e⃗_{i}, …, e⃗_{j} − e⃗_{i}, …, −e⃗_{i}, …, e⃗_{λ} − e⃗_{i}

(V_{j})  e⃗_{0}, …, e⃗_{j}, …, e⃗_{i}, …, e⃗_{λ}

5. δ-pregeometry and red extensions

Let L^{morley} be the language which consist of 0, +, −, and a relation symbol for every quantifier free L–formula. 𝕌 becomes an L^{morley}–structure in a natural way. We extend this language to L^{∗} = L^{morley}∪{R}, where R is a unary predicate, which will yield the red coloring. The L^{∗}-structures A we consider are additive subgroups of 𝕌 with a distinguished additive subgroup R(A) (viewed as an 𝔽_{p}-vector space). The elements of R(A) are the red points of A, the others white. We write A ⊂ B if in the extended language we also have that R(A) = R(B) ∩ A. We consider the following function, as introduced by B. Poizat in [11]:

δ(A) = 2 trdeg A − dim R(A)

for finite A.

Note that δ satisfy the following (cf. [3]):

(1) δ(0) = 0,
(2) δ⟨A, B⟩ + δ(A ∩ B) ≤ δ(A) + δ(B).

If dim(R(A)/R(B)) is finite^{4}, we define the relative δ-value of A over B by:

δ(A/B) = 2 trdeg(A/B) − dim(R(A)/R(B)).

If also B is finite, we have

δ(A/B) = δ⟨A, B⟩ − δ(B).

We will later make also use of the notation δ(a⃗/B) for δ(⟨a⃗⟩/B).

4We do not assume that B is included in A.

Following [7], we say that Y is self-sufficient or strong in X (denoted as Y ≤ X) if for all finite A ⊂ X, we have that δ(A/Y) ≥ 0. It is easy^{5} to see that selfsufficiency is transitive. Moreover, the intersection of self-sufficient subsets of X is again self-sufficient and each subset S of X is contained in a smallest self-sufficient subset, its self-sufficient closure cl_{X}(S).

A proper extension Y ≤ Z is minimal if no Y ⊊ Y′ ⊊ Z is self-sufficient in Z. The set Z \ Y must be finite, which allows us to express minimality by

δ(Z/Y′) < 0 for all Y ⊊ Y′ ⊊ Z.

We state a basic fact (whose proof is similar as in Lemma 4.1 in [3]):

Lemma 5.1. Let B ≤ A be a minimal extension. We have one of the following cases:

(1) A = ⟨B, a⟩ for a white element a and R(A) = R(B). Moreover, δ(A/B) = 0 or 2, depending whether a is algebraic or transcendental over B.

(2) A = ⟨B, a⃗⟩ for a basis a⃗ = (a_{1}, …, a_{n}) of R(A) over B. We have two subcases:

(a) n ≥ 2, δ(A/B) = 0 and a⃗ is a generic element of a prealgebraic X ⊂ 𝕌^{n} defined over acl(B). In this case we call A a minimal prealgebraic extension of B.

(b) n = 1, δ(A/B) = 1 and a_{1} is transcendental over B.

Note that in (2) all new elements of A are transcendental over B. We call such an extension transcendental.

Proof. First assume that all red elements in A are in B. Let a be in A \ B. Then ⟨B, a⟩ is self-sufficient in A and by minimality it is A. If A is algebraic over B, then δ(A/B) = 0. Otherwise δ(A/B) = 2. So we are in case (1).

Otherwise choose a basis a⃗ = (a_{1}, …, a_{n}) of R(A) over B and set A′ = ⟨B, a⃗⟩. Then A′ ≤ A and by minimality A′ = A: We are in case (2). There are two subcases:

(2a): There is a proper extension C of B in A such that δ(C/B) = 0. By minimality we have C = A. Then n = 2k where k is the transcendence degree of A over B. By minimality we have 2 · trdeg(a⃗/U, B) < n − dim(U) for all non–trivial subspaces U of ⟨a⃗⟩. Choose an acl(B)–definable set X of Morley rank k and Morley degree 1, which contains a⃗. Then a⃗ is generic in X and X is prealgebraic.

(2b): Otherwise every extension A′′ of B in A with δ(A′′/B) = 1 is self-sufficient in A. Hence we get A = ⟨B, a_{1}⟩ and δ(A/B) = 1. □

Theorem 4.4 now yields the following result:

Lemma 5.2. Let B ≤ A minimal prealgebraic extension of B. Then there is a unique code α and parameters b⃗ in acl(B) such that A is generated over B by a generic red realization of φ_{α}(x⃗, b⃗). □

The final lemma of this section can be proved as Lemma 3A in [6] (see Lemma 4.5 in [3]). We will all throughout this paper use it in order to show that a realization of a code is generic over some set of parameters.

5Actually the following facts are formal consequences of (1) and (2).

Lemma 5.3. Let α be a code, b⃗ ∈ acl(B) and a⃗ be a red realization of φ_{α}(x⃗, b⃗) which does not completely lie in B. Then, the following holds:

(1) δ(a⃗/B) ≤ 0

(2) If δ(a⃗/B) = 0, a⃗ is a generic realization of φ_{α}(x⃗, b⃗). In particular a⃗ is 𝔽_{p}– linearly independent over acl(B). □

6. A counting result

Definition 6.1. Let K be the class of all L^{∗}-structures M (i.e colored subspaces of 𝕌) such that ∅ ≤ M.

So M ∈ K iff δ(A) ≥ 0 for all finite subsets A of M. It is easy to see that K can be described by a set of universal sentences.^{6}

If M belongs to K the self-sufficient closure of a finite subset of M is finite.

Definition 6.2. In K we define that M′ is an amalgam of M and A over B, if • M and A are self-sufficient in M′,

• M and A are algebraically independent over B,

• M′ = ⟨M, A⟩. If in addition

• M ∩ A = B, we call M′ a free amalgam.

Using standard arguments (cf.[10]) one shows:

Lemma 6.3. If M, B, and A are in K and B ≤ M and B ≤ A, there is an amalgam M′ of M and A over B. If M or A are transcendental over B, we can find a free amalgam M′. □

Definition 6.4. For structures in K a difference sequence for α ∈ C of length λ is a red realization e⃗_{0}, …, e⃗_{λ} of Ψ_{α}(x⃗_{0}, …, x⃗_{λ}).

The next Lemma is the key tool in order to classify the structures which will be amalgamated with Fraïssé’s method. This characterization will be useful for exhibiting the theory of the Fraïssé limit to be obtained in Section 8.

Lemma 6.5. Given a code α and a natural number r, there is some λ(r, α) = λ ≥ 0 such that for every strong extension M ≤ N and every difference sequence e⃗_{0}, …, e⃗_{μ} in N, with canonical parameter b⃗ and λ ≤ μ, either

• the canonical parameter of some λ-derived sequence of e⃗_{0}, …, e⃗_{μ} lies in

dcl(M), or

• the sequence e⃗_{0}, …, e⃗_{λ} contains a Morley sequence of φ_{α}(x⃗, b⃗) over M of

length r.

By “Morley sequence” we mean Morley sequence in the sense of ACF_{p}.

6Her we use again the finiteness of 𝔽_{p}.

Proof. If there are more than m_{α} many of the e⃗_{i} in the same class of R(N)^{n_{α}}/R(M)^{n_{α}}, we subtract one of these elements from the others and obtain a derived sequence with m_{α} many elements in M, whose canonical base lies in dcl(M). So we assume that each class of N^{n_{α}}/M^{n_{α}} contains at most m_{α} many e⃗_{i}. Work over M′ = M ∪ {e⃗_{0}, …, e⃗_{m_{α}−1}} and observe that b⃗ ∈ dcl(M′). All we need now from the sequence (e⃗_{i}) is that the e⃗_{i} realize φ_{α}(x⃗, b⃗).

Set

s = dim(e⃗_{0}, …, e⃗_{λ}/⟨M′⟩).

Then dim(e⃗_{0}, …, e⃗_{λ}/M) ≤ s + m_{α}n_{α}. Thus by our assumption

λ + 1 ≤ m_{α} p^{(s+m_{α}n_{α})n_{α}}.

Consider the following sets of indices:

X_{1} = {i ≤ λ | trdeg(e⃗_{i}/M′, e⃗_{0}, …, e⃗_{i−1}) = k_{α}},
X_{2} = {i ∉ X_{1} | dim(e⃗_{i}/⟨M′, e⃗_{0}, …, e⃗_{i−1}⟩) > 0}.

It is clear that

s ≤ |X_{1}| n_{α} + |X_{2}| n_{α}.

With the notation δ(i) = δ(e⃗_{i}/⟨M′, e⃗_{0}, …, e⃗_{i−1}⟩), Lemma 5.3 implies that δ(i) < 0 if x ∈ X_{2}, and δ(i) = 0 otherwise. Since M ≤ N we have

0 ≤ δ(e⃗_{0}, …, e⃗_{λ}/M) = δ(e⃗_{0}, …, e⃗_{m_{α}−1}/M) + ∑_{i=1}^{λ} δ(i) ≤ m_{α}n_{α} − |X_{2}|.

If we put the three inequalities together, we obtain

λ + 1 ≤ m_{α} p^{((|X_{1}|+m_{α}n_{α})n_{α}+m_{α}n_{α})n_{α}}.

If λ is large enough, |X_{1}| ≥ r and (e⃗_{i})_{i∈X_{1}} is our Morley sequence. □

7. Leaving K^{μ}

We choose two finite-to-one functions μ^{∗} and μ defined on C with values on ℕ such that the following inequalities hold:

• μ(α) ≥ m_{α}

• μ^{∗}(α) ≥ max(λ(m_{α} + 1, α) + 1, n_{α} + 1)
• μ(α) ≥ λ(μ^{∗}(α), α) + 1

Definition 7.1. The class K^{μ} is the class of all L^{∗}-structures M (i.e colored subspaces of 𝕌) such that:

• ∅ ≤ M.

• No α in C has a difference sequence in M of length μ(α) + 1.

It is easy to see that K^{μ}, as K, is axiomatizable by universal L^{∗}–sentences. In fact, we have

Remark 7.2. For all α there is a universal sentence θ_{α}, which is true in M ∈ K iff M has no difference sequence for α of length μ(α) + 1.

All models of the L^{∗}-theory T^{μ} we intend to construct will be in K^{μ}. We want that as many copies as possible of a prealgebraic extension of a strong subset of a model are realized. In this section we show that this is an elementary property of the model. The results are also important for the amalgamation in K^{μ}.

Lemma 7.3. Assume M and M′ are structures in K. M′ is a prealgebraic minimal extension of M, M is in K^{μ} and M′ is not in K^{μ}. Let e⃗_{0}, …, e⃗_{μ(α)} be a difference sequence for a good code α in M′, such that its canonical parameter c⃗ is in acl(M).^{7}

Then we find such a difference sequence d⃗_{0}, …, d⃗_{μ(α)} for α in M′ with the same canonical parameter such that d⃗_{0}, …, d⃗_{μ(α)−1} are in M, d⃗_{μ(α)} is an M-generic realization of φ_{α}(x⃗, c⃗) and generates M′ over M.

If we cannot find the new sequence by a permutation of the old one, then α is a group code and the new sequence is obtained using operations as e⃗_{j} is replaced by some He⃗_{j} − e⃗_{i} where H is in Inv(φ_{α}(x⃗, c⃗)).

Furthermore α is the unique good code that describes M′ over M.

Proof. Since M is in K^{μ}, there is some e⃗_{i} not completely contained in M. Since M is strong in M′ by Lemma 5.3 e⃗_{i} is M-generic. Since M′ is minimal over M it generates M′ over M. If there is some other e⃗_{j} not completely contained in M, then again e⃗_{j} is M-generic and generates M′ over M. Hence e⃗_{i} = He⃗_{j} − m⃗_{j} where H is GL_{n_{α}}(𝔽_{p}) and m⃗_{j} is in M. Then He⃗_{j} − e⃗_{i} is in M. Since e⃗_{j} is M-generic, we have

e⃗_{j} | ⌣_{c⃗} He⃗_{j} − e⃗_{i}.

By the properties of a difference sequence it follows that α is a group code and H is in Inv(φ_{α}(x⃗, c⃗)). If we replace e⃗_{j} by He⃗_{j} − e⃗_{i}, then we obtain again a difference sequence with the same canonical parameter and this sequence has one more element in M. We can iterate the argument to obtain the assertion.

Finally every prealgebraic set that gives us M′ over M determines a unique code by Theorem 4.4. All such prealgebraic sets can be transformed into each other using GL_{n_{α}}(𝔽_{p}) and translations. Hence there is a unique good code for this situation. □

Corollary 7.4. Let M be in K^{μ} and M ≤ M′ a minimal extension. If M′ has linear dimension 1 over M, then M′ is in K^{μ}.

Otherwise, in the prealgebraic case, M′ is in K^{μ} if and only if none of the following two conditions holds:

a) There is a code α ∈ C and a difference sequence e⃗_{0}, …, e⃗_{μ(α)} for α in M′

such that:

(i) e⃗_{0}, …, e⃗_{μ(α)−1} are contained in M.

(ii) M′ = ⟨M, e⃗_{μ(α)}⟩.

(iii) In this case α is the unique good code that describes M′ over M.

b) There exists a code α ∈ C and a difference sequence for α in M′ of length μ(α) + 1 with canonical parameter b⃗ with μ^{∗}(α) many elements which form a Morley sequence of φ_{α}(x⃗, b⃗) over M.

Proof. Consider first the case where dim(M′/M) = 1. If R(M′) = R(M), then M′ does not contain new difference sequences, so M′ is in K^{μ}. Now assume that dim(R(M′)/M) = 1, and that M′ is not in K^{μ} witnessed by some difference sequence e⃗_{0}, …, e⃗_{μ(α)}. If the canonical parameter b⃗ lies in dcl(M), since no e⃗_{i} can be linearly independent over M, all e_{i} would be in M, which is not possible. Therefore, since μ(α) ≥ λ(1, α)^{8} and M ≤ M′, some e⃗_{j} is an M–generic realization of φ_{α}(x⃗, b⃗), impossible.

7The Lemma implies c ∈ dcl(M).

Finally, let M′/M be prealgebraic. Clearly, if a) or b) hold, then M′ is not in K^{μ}. Assume, for the converse, that M′ is not in K^{μ}, witnessed by some good code α and a difference sequence {e⃗_{i}}_{0≤i≤μ(α)} with canonical parameter b⃗. First, suppose that the canonical parameter of some derived difference sequence lies in acl(M). Then Lemma 7.3 yields case a) as desired.

Otherwise, since μ(α) ≥ λ(μ^{∗}(α), α), {e⃗_{i}}_{0≤i≤μ(α)} contains a Morley sequence of φ_{α}(x⃗, b⃗) over M of length μ^{∗}(α). Therefore, b) holds. □

Corollary 7.5. For each good code α there is an ∀∃–sentence χ_{α} such that a structure M in K^{μ} satisfies χ_{α} iff M has no prealgebraic minimal extension in K^{μ}, which is given by α.

Proof. Let M′ = ⟨M, a⃗⟩, where a⃗ is a red M–generic realization of φ_{α}(x⃗, b⃗), for some b⃗ ∈ M. If M′ is not in K^{μ}, we are in the cases a) or b) of Corollary 7.4. In case a) M′ contains a difference sequence of length μ(α) + 1 for α, in case b) difference sequence of length μ(β) + 1 for a good code β, which contains a subsequence of length μ^{∗}(β), linearly independent over M, so μ^{∗}(β)n_{β} ≤ n_{α}. Since μ^{∗} is finite-to-1, only a finite set C_{α} of codes β can occur.

Set C′_{α} = C_{α} ∪{α}. Then M has no prealgebraic minimal extension in K^{μ}, given by α iff

∀b⃗ ∈ M, ∀ red M–generic realizations a⃗ of φ_{α}(x⃗, b⃗) ⋁_{β∈C′_{α}}⟨M, a⃗⟩ ⊧ ¬ θ_{β}.

Since the L–type of a⃗ over M is uniformly definable over b⃗, this can easily expressed by an ∀∃–sentence. □

8. The Fraïssé model

We show now that the class K^{μ} has the Amalgamation Property, and hence, we can obtain rich fields as introduced by Poizat in [10].

The following lemma replaces an easy argument in the case of black points by a more sophisticated one for our purposes.

Lemma 8.1. Let A, B, M be structures in K where B is a strong substructure of M and of A. Let M′ be a free amalgam of M and A over B. Assume that e⃗_{0}, …, e⃗_{μ(α)} is a difference sequence for a good code α in M′.

Then there is a derived difference sequence of the above sequence with the canonical parameter in acl(M) or in acl(A).

Proof. We assume that the assertion of the lemma is not true. Let b⃗ be the canonical parameter of the sequence. If we apply Lemma 6.5, then, since μ^{∗}(α) ≥ λ(m_{α} + 1, α) + 1 we get a subsequence with more than λ(m_{α} + 1, α) elements that form a Morley sequence of φ_{α}(x⃗, b⃗) over M. From this again we get a subsequence of length m_{α} + 1 that is a Morley sequence over M and also over A. Write these elements as e⃗_{i} for 0 ≤ i ≤ m_{α}. If E = {e⃗_{0}, …, e⃗_{m_{α}−1}}, we have that b⃗ ∈ dcl(E) and

8We silently assume that λ is monotonous in the first argument.

e⃗_{m_{α}} | ⌣_{b⃗} M, E,  e⃗_{m_{α}} | ⌣_{b⃗} A, E.

Write every element of E as the sum of a tuple in M and a tuple in A. Define E_{M} to be the set of all elements of M which occur, E_{A} similarly and set E′ = E_{M} ∪E_{A}. Then also b⃗ ∈ dcl(E′) and, since E′ and E are interdefinable over M and over A, we have

e⃗_{m_{α}} | ⌣_{b⃗} M, E′,  e⃗_{m_{α}} | ⌣_{b⃗} A, E′.

Whence

e⃗_{m_{α}} | ⌣_{BE′} M,  e⃗_{m_{α}} | ⌣_{BE′} A.

Furthermore

M | ⌣_{BE′} A.

Write e⃗_{m_{α}} = m⃗ + a⃗ for m⃗ ∈ M and a⃗ ∈ A. Then {e⃗_{m_{α}}, m⃗, a⃗} is a pairwise independent triple over B, E′. Whence φ_{α}(x⃗, b⃗) is a group formula for a definable group G and b⃗ is the canonical parameter of G. Now, a⃗ is a generic element of an acl(B, E′)–definable coset of G and b⃗ is definable from the canonical base of p = tp(a⃗/ acl(B, E′)). Note that

a⃗ | ⌣_{BE_{A}} E′

so the canonical base of p is in acl(A), and we have b ∈ acl(A). This contradicts our assumption. □

A self-sufficient embedding of B in A is an isomorphism of B onto a self-sufficient subset of A.

Theorem 8.2. The class K^{μ} has the amalgamation property with respect to selfsufficient embeddings.

Proof. Let B ≤ M and B ≤ A be structures in K^{μ}. We need to show that there is an extension M′ of M in K^{μ}, with M ≤ M′ and some B ≤ A′ ≤ M′ such that A and A′ are isomorphic over B. We may assume that both extensions B ≤ A and B ≤ M are minimal, by splitting them into minimal ones.

Case 1. dim(M/B) = 1 or dim(A/B) = 1. By Lemma 6.3 M and A can be strongly embedded in an amalgam M′. Since dim(M′/A) ≤ 1 or dim(M′/M) ≤ 1, Corollary 7.4 implies that M′ ∈ K^{μ}.

Case 2. Both extensions M/B and A/B are prealgebraic. Let M′ be a free amalgam of M and A over B, which exists by Lemma 6.3. We are done if M′ belongs to K^{μ}. So assume that M′ does not belong to K^{μ}. We need then show that M and A are isomorphic over B.

There is a good code α with a difference sequence e⃗_{0}, …, e⃗_{μ(α)} in M′. By Lemma 8.1 and symmetry we may assume that its canonical parameter b⃗ lies in acl(M). By Lemma 7.3 we may assume that e⃗_{0}, …, e⃗_{μ(α)−1} are in M and e⃗_{μ(α)} is an M–generic realization of φ_{α}(x⃗, b⃗) which generates M′ over M.

Subcase 2.a We first assume that the canonical parameter for some (μ(α) − 1)- derived difference sequence, is in dcl(B). Since this difference sequence has the same properties, we denote it again by {e⃗_{i}}_{0≤i≤μ(α)}. By Lemma 5.3 there are two cases:

Subcase 2.a.1 e⃗_{μ(α)} is in A. By minimality of A over B, we have that A = ⟨B, e⃗_{μ(α)}⟩. Since A is in K^{μ}, there exists an e⃗_{i} which lies in M but not in B. It follows from B ≤ M and Lemma 5.3 that e⃗_{i} is B-generic, and isomorphic to A over B. This shows that M and A are isomorphic over B.

Subcase 2.a.2 e⃗_{μ(α)} is an A-generic realization of φ_{α}(x⃗, b⃗). Write e⃗_{μ(α)} = m⃗ + a⃗, and note that 0 = δ(e⃗_{μ(α)}/M) = δ(a⃗/M) = δ(a⃗/B), so a⃗ generates A over B (again by minimality of B ≤ A). Moreover, {e⃗_{μ(α)}, m⃗, a⃗} is a B-independent triple. By Note 2.2, −m⃗ and a⃗ have the same type over B and hence we obtain the desired isomorphism.

Subcase 2.b No (μ(α) − 1)-derived difference sequence has canonical basis c⃗ in dcl(B). Write e⃗_{μ(α)} = m⃗ + a⃗ (as above, a⃗ generates A over B). Note that only n_{α} many elements of a Morley sequence may fork with a fixed tuple of length n_{α}, so Lemma 6.5 (applied to B ≤ M′) gives us an e⃗_{i} ∈ M which is an B, m⃗–generic realization of φ_{α}(x⃗, c⃗) since μ^{∗}(α) ≥ n_{α} + 1. In this case, e⃗_{i} − m⃗ is B-isomorphic to a⃗. □

We call M in K^{μ} rich if for any B ≤ M finite and any finite extension B ≤ A of members of K^{μ}, there is a self-sufficient substructure A′ ≤ M with B ≤ A′ and B-isomorphic to A.

Corollary 8.3. There is a unique (up to isomorphism) countable rich structure M in K^{μ}. All rich structures are L_{∞,ω}^{∗}-equivalent.

9. Axiomatization

In this section we will show that rich fields in K^{μ} are exactly the ω-saturated models of a specific theory T^{μ}. We work in the extended language L^{∗}.

Let T^{μ} denote following axiom schemes:

Universal Axioms:

(1) Any model is a member of K^{μ}. ∀∃ Axioms:

(2) Any model is an algebraically closed field of characteristic p.

(3) Given a good code α and b⃗ in the model, the extension of M generated by a M-generic red realization of φ_{α}(x⃗, b⃗) does not belong to K^{μ}.

Corollary 7.5 shows that axiom (3) is elementary. The following result gives us an equivalent description of a rich model, and moreover, yields an explicit axiomatization of its theory.

Theorem 9.1. An L^{∗}-structure is rich if and only if it is an ω-saturated model of T^{μ}.

Proof. The argument of the proof is similar to the one in [3] based on ideas from [10]. We first show that ω-saturated models of T are rich, and also that a rich model satisfies all axioms. Hence, by the above and ∞-equivalence of rich models, we obtain the other implication.

Let M ⊧ T^{μ} be ω-saturated. Let B ≤ M and B ≤ A be finite L^{∗}-structures in K^{μ}. We may assume that B ≤ A is minimal. We can distinguish four different cases, as given by Lemma 5.1:

A/B is algebraic: We are done by Axiom (2).

A/B is prealgebraic: Consider the free amalgam M′ of M and A over B. In this case A is given over B by a red generic realization of a good code. Axiom (3) yields that M′ does not belong to K^{μ}. By Theorem 8.2 A has a strong embedding over B into M.

A is generated over B by a red element that is transcendental over B: We will approximate this extension by prealgebraic minimal extensions A_{n} = ⟨B, x_{1}, x_{2}⟩, where the x_{i} are red, transcendental over B and x_{1}^{n} = x_{2}. The sequence A_{n}/B converges (in the space of colored L-types) to the extension A_{∞}/B, where A_{∞} = ⟨B, a_{1}, a_{2}⟩, with a_{1}, a_{2} red and algebraically independent over B. This extension decomposes into B ≤ ⟨B, a_{1}⟩ ≤ A_{∞}, which implies by Corollary 7.4 that A_{∞} belongs to K^{μ}. So, since μ is finite to one, A_{n} belongs to K^{μ} for large n. As shown above, we can find self-sufficient B-copies of A_{n} in M for large n. By saturation of M, A_{∞} is also self-sufficiently embeddable over B. Since A is isomorphic to ⟨B, a_{1}⟩, we conclude that there is a self-sufficient B-copy of A in M.

A is generated over B by a white element that is transcendental over B and R(A) = R(B): Consider for each n the extension B ≤ ⟨B, z_{1}⟩ ≤ ⟨B, z_{1}, z_{2}⟩ = C_{n}, where z_{1} is red, z_{1}^{n} = z_{2} is white and R(C_{n}) = ⟨R(B), z_{1}⟩. By the above each C_{n} can be strongly embedded in M. Therefore also the limit C_{∞} = ⟨B, c_{1}, c_{2}⟩ can be strongly embedded in M. Since ⟨C, c_{2}⟩ ≤ C_{∞} is B–isomorphic to A, we are done.

Suppose now that M is a rich field. We first show that M is algebraically closed. Let a ∈ acl(M). Choose a finite set B in M such that a is in acl(B). Taking the self-sufficient closure of B in M, we can assume that B ≤ M. Paint ⟨B, a⟩ \ B in white. It is clear that ⟨B, a⟩ is in K^{μ} (since B is) and B ≤ ⟨B, a⟩. By richness, we find a copy of a in M over B. This yields (2).

For Axiom (3), let α and b⃗ be as in the statement such that ⟨M, a⃗⟩ is in K^{μ}, where a⃗ is some generic red realization of φ_{α}(x⃗, b⃗) over M. We show a contradiction. Choose some set B ≤ M containing b⃗. Again, B ≤ ⟨B, a⃗⟩, and by richness, we get a B-copy of a⃗ in M, say a′. Take now some finite C ≤ M containing B ∪ a′. We have that C ≤ ⟨C, a⃗⟩. We can iterate and obtain a Morley sequence in M for φ_{α}(x⃗, b⃗) of length μ(α) + 2. This yields a difference sequence for α of length μ(α) + 1 in M by property (a) of Ψ_{α}. □

Corollary 9.2. T^{μ} is complete. Two tuples a⃗ and a′ in two models M and M′ have the same type iff there is an isomorphism f : cl(a⃗) → cl(a′) which maps a⃗ to a′.

Proof. Corollary 8.3 and Theorem 9.1 provides us a rich model M^{∗} of T^{μ}. Let M be any model of T. By theorem 9.1 there is a rich M′ ≡ M. So M′ ≡_{∞,ω} M^{∗}, which proves completeness.

To prove the second statement choose ω–saturated elementary extensions M ≺ N and M′ ≺ N′ . By Corollary 9.3 (we use only the part which does not rely on 9.2.) is M ≤ N and M′ ≤ N′, so “cl” does not change.

An isomorphism f : cl(a⃗) → cl(a′) is now part of a back-and-forth-system of partial isomorphisms. Whence f is a elementary map.

For the converse suppose that a⃗ and a′ have the same type. There is an elementary map f : cl(a⃗) → M′ which maps a⃗ onto a′. We write A′ for f(cl(a⃗)). Since A′ has the same type as cl(a⃗), A′ is self-sufficient in M, and we can conclude that A′ = cl(a′).

□

Corollary 9.3. Let M ⊂ N be an extension of two models of T. Then M ≺ N iff M ≤ N.

Proof. If M ≰ N, there is an a ∈ N with δ(a/M) < 0. We find a finite B ≤ M with δ(a/B) < 0. This is witnessed by the truth of an L_{1} ∪ L_{2}–formula φ(a, b⃗). φ(x, b⃗) is not satisfiable in M, whence M ⊀ N.

The converse M ≤ N ⇒ M ≺ N follows directly from 9.2, since M ≤ N ensures that cl(a) is the same in M and N. □

10. Rank computations

In this section we show that T^{μ} has Morley rank 2. First we describe algebraic closure acl^{∗} in models M of T^{μ}.

Definition 10.1. Let M ⊧ T^{μ}, B a subset of M. Then cl_{d}(B) is the union of all finite A ⊂ M with δ(A/ cl(B)) = 0.

It is easy to see that

cl_{d}(B) = {a ∈ M | d(a/B) = 0},

where we use the notation

d(A/B) = δ(cl⟨A, B⟩/ cl(B)).

Lemma 10.2. Both closures acl^{∗} and cl_{d} agree on models M of T^{μ}.

Proof. We may assume that M is ω-saturated by Lemma 9.3. If B is finite, then so is cl(B), hence contained in acl^{∗}(B). So we may assume that B is finite and self-sufficient in M.

First we show cl_{d} is part of acl^{∗}. We consider δ(A/ cl(B)) = 0. We decompose the extension A/B into a sequence of minimal extensions and use induction on the length of the sequence. Hence we have to consider A/B minimal δ(A/B) = 0 with B ≤ M. By Lemma 5.1 two cases may arise:

Case 1) A is algebraic over B in the field sense. But then A ⊆ acl^{∗}(B).

Case 2) A is a prealgebraic extension of B.

By Theorem 4.4 there are a good code α and parameters b⃗ in acl(B) such that A = ⟨B, a⃗⟩ where a⃗ is a generic red solution of φ_{α}(x⃗, b⃗). We show that φ_{α}(x⃗, b⃗) has only finitely many red solutions, which implies a⃗ ∈ acl^{∗}(B) and therefore A ⊂ acl^{∗}(B). Assume that there are infinitely many solutions. Then there is a solution e⃗_{0} which is not contained in B. By Lemma 5.3 e⃗_{0} is a B-generic realization of φ_{α}(x⃗, b⃗). We have δ(e⃗_{0}/B) = 0, and therefore B′ = ⟨B, e⃗_{0}⟩ ≤ M. Since B′ is again finite, we can find a B′–generic solution e⃗_{1}, etc. So we find an infinite Morley sequence (e⃗_{i}) for φ_{α}(x⃗, b⃗) in M. As above, this contradicts Axiom (1).

Now consider an element a outside of cl_{d}(B). If A = cl(B, a), we have δ(A/B) > 0. Decompose A/B into minimal extensions:

B = A_{0} ≤ A_{1} ≤ ⋯ ≤ A_{n} = A

There is some i < n with δ(A_{i+1}/A_{i}) > 0, which implies that dim(A_{i+1}/A_{i}) = 1 by 5.1. We saw in 7.4 that free amalgams with transcendental strong extensions of dimension 1 cannot leave K^{μ}. So, since M is rich, there are infinitely many A′ ≤ M which are over A_{i} isomorphic to A_{i+1}. By 9.2 they have all the same type over A_{i}. Hence A_{i+1} is not ∗–algebraic over A_{i}. Since A_{i+1} is algebraic over Ba, it follows that a ∉ acl^{∗}(B). □

Theorem 10.3. T^{μ} has Morley rank 2. In particular, the generic type has rank 2 and the generic red type has rank 1. The algebraic closure in models of T^{μ} is cl_{d}.

Proof. We consider an ω-saturated model M of T^{μ} inside a monster model. We compute MR^{∗}(a/M) for elements a in the monster model.

We have

0 ≤ d(a/M) ≤ δ(a/M) ≤ 2.

So there are three cases:

d(a/M) = 0: Then a is algebraic over M, i.e. a ∈ M and we have MR^{∗}(a/M) = 0.

d(a/M) = 1: If a is red, it follows that ⟨M, a⟩ is self–sufficient. So tp^{∗}(a/M) is uniquely determined. Since all other red types over M are algebraic, we conclude that MR^{∗}(a/M) = 1 and the red subgroup is strongly minimal. If a is not red, then cl(M, a) contains a red element c outside M. If follows ⟨M, c⟩ ≤ cl(M, a) and d(a/M, c) = 0. This implies that a and c are L^{∗}–interalgebraic over M and MR^{∗}(a/M) = MR^{∗}(c/M) = 1.

d(a/M) = 2: Then ⟨M, a⟩ is self–sufficient. So tp^{∗}(a/M) is uniquely determined, since the isomorphism type of of ⟨M, a⟩ is determined by R(⟨M, a⟩) = R(M). Since all other types over M have Morley rank ≤ 1, we have MR^{∗}(a/M) ≤ 2. But R(M) is an infinite subgroup of M of infinite index. Hence T^{μ} has Morley rank 2 and MR^{∗}(a/M) = 2. □

Remark 10.4. We note the following without proof.

(1) As any complete theory of fields of finite Morley rank T^{μ} is ω_{1}–categorical.

(2) Lindström’s theorem implies that T^{μ} is model complete. This can be directly proved as in [4].

(3) For every natural number r ≥ 2 there is a theory of a field of Morley rank r with an additive red subgroup of Morley rank r − 1. This can be proved as in [3] using

δ(A) = r · trdeg(A) − dim(A).

(4) In models (K, R) of T^{μ}, we have

𝔽_{p} = {x ∈ K | x · R ⊂ R}.

If we consider 𝔽_{p^{n}}–vector spaces R and use 𝔽_{p^{n}}–linear dimension, we can produce a field (K, R) of Morley rank 2, where

𝔽_{p^{n}} = {x ∈ K | x · R ⊂ R}.

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   0608433).

Institut für Mathematik, Humboldt-Universität zu Berlin, D-10099 Berlin, Germany

  E-mail address: baudisch@mathematik.hu-berlin.de

Institut für Mathematik, Humboldt-Universität zu Berlin, D-10099 Berlin, Germany

Institut Camille Jordan, Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne cedex, France

  E-mail address: pizarro@mathematik.hu-berlin.de

Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, D-79104 Freiburg, Germany

  E-mail address: ziegler@uni-freiburg.de