ON FIELDS AND COLORS

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

Abstract. We exhibit a simplified version of the construction of a field of Morley rank p with a predicate of rank p − 1, extracting the main ideas for the construction from previous papers and refining the arguments. Moreover, an explicit axiomatization is given, and ranks are computed.

1. Introduction

Zil’ber posed the question whether or not every strongly minimal set whose geometry was not locally modular arose from an algebraic curve over an algebraically closed field. The conjecture, true in the case of Zariski Geometries [8], remained open until E. Hrushovski [7] refuted it developing a procedure, taking ideas from Fraïssé, in order to construct countable structures with a richer and more complicated geometry starting from simpler ones. Moreover, he was able to merge two algebraically closed fields of different characteristics into one strongly minimal set [6]. This procedure was later adapted by Poizat [11] to obtain an algebraically closed field of any given characteristic with a predicate (whose elements were called black, after some considerations on the political correctness of such a choice of terminology) such that the field has Morley rank ω2 and the black points ω. He then used Hrushovski’s collapsing method and produced “rich” fields of rank 2 with black points of rank 1, provided the rich field is ω–saturated. A proof of ω–saturation was supplied by Baldwin and Holland ([1]). Poizat and Baldwin & Holland also explained how to obtain fields of rank p with a predicate of rank 1 and p − 1, respectively.

The main goal of this work is to give a complete self-contained proof of the above facts simplifying as much as possible the arguments. One of the novelties of this work is exhibiting an axiomatization for the resulting theory, obtained by direct translation of Hrushovski’s fusion article [6] to the case of colored fields. Actually, we use a simplified version (see [12]) of the aforementioned article, following the spirit of Poizat’s black points.

All throughout this work a saturated enough algebraically closed field ℂ of some given characteristic q and a natural number p ≥ 2. We will prove the following:

Date: January 14, 2005.

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

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

A. Martin-Pizarro conducted research supported by the DFG under Forschungsstipendium MA3310/1-1

��������� ������ ������ �������� ���������� ������ ������������ ����� ���������� � �������� Main Theorem ([11],[1]). ℂ has a subset N such that (ℂ, N) has Morley rank p and N has Morley rank p − 1.

This paper is structured as follows: We first consider finite partial substructures of ℂ with some points colored in black. A δ function is introduced, and Hrushovski’s codes [6] are used to described minimal extension (with a small correction from their original definition). The number of certain such extensions is bounded with a μ function. In this case, we can proceed with the collapse, and the resulting structure is a rich field as in [11]. We show that rich fields are exactly the ω-saturated models of a given theory, whose axioms are explictly given. Finally, we compute the Morley rank in terms of δ.

This work originated from a seminar held at Humboldt-Universität zu Berlin directed by the first and third author [3] during 2003–2004 in which the second author took part. We would specially like to thank Juan R. Bueno for his help in a preliminary version of this article [4] during his stay in Berlin.

2. Codes

In this section, we work exclusively inside ℂ. All formulae are L–formulas, where L is the ring language

Definition 2.1. A code α is a tuple consisting of the following objects: Natural numbers n_{α}, m_{α}, k_{α} and formulae φ_{α}(x⃗, y⃗) and ψ_{α}(x⃗_{1}, …, x⃗_{m_{α}}, y⃗) such that the following holds (We will write θ_{α}(y⃗) = ∃x⃗ φ_{α}(x⃗, y⃗)):

(i) length(x⃗) = length(x⃗_{i}) = n_{α} (ii) If ⊧ θ_{α}(b⃗), then φ_{α}(x⃗, b⃗) has Morley rank k_{α} and degree 1. (iii) Let a⃗ ⊧ φ_{α}(x⃗, b⃗) be generic. For s ⊂ {1, …, n_{α}}, write a_{s} = {a_{j}}_{j∈s}. Then,

for every i ≤ n_{α} and a′ ⊧ φ_{α}(x⃗, b⃗′), we have that:

\begin{aligned}a_{i} ∈ acl(a_{s}, b⃗) & ⟹ & a′_{i} ∈ acl(a′_{s}, b⃗′) \\ a_{i} ∈ a_{s}b⃗ & ⟹ & a′_{i} ∈ a′_{s}b⃗′ \\ a_{i} ∉ a_{s}b⃗ & ⟹ & a′_{i} ∉ a′_{s}b⃗′ \end{aligned}

(iv) If ⊧ θ_{α}(b⃗), then MR(φ_{α}(x⃗, b⃗) △ φ_{α}(x⃗, b⃗′)) < k_{α} ⟹ b⃗ = b⃗′.

(v) ⊧ ψ_{α}(x⃗_{1}, …, x⃗_{m_{α}}, b⃗) implies that b⃗ ∈ dcl(x⃗_{1}, …, x⃗_{m_{α}}).^{1} (vi) ψ_{α}(x⃗_{1}, …, x⃗_{m_{α}}, b⃗) is consistent for all b⃗ ⊧ θ_{α}(y⃗). (vii) Given (a⃗_{1}, …, a⃗_{m_{α}}, b⃗) ⊧ ψ_{α} and a generic a′ realizing φ_{α}(x⃗, b⃗) and indepen-

dent from a⃗_{1}, …, a⃗_{m_{α}} over b⃗, it follows that

⋀_{i=1}^{m_{α}} ψ_{α}(a⃗_{1}, a⃗_{2}, …, a⃗_{i−1}, a′, a⃗_{i+1}, …, a⃗_{m_{α}}, b⃗)

holds.

Lemma 2.2. If ⊧ θ_{α}(b⃗), we have that b⃗ is a canonical basis of the type of Morley rank k_{α} determined by φ_{α}(x⃗, b⃗).

Proof. This follows immediately from (iv). □

1Note that the definable closure dcl(x⃗) is the perfect hull of the field generated by x⃗.

Lemma 2.3. For each definable set X of n-tuples of Morley rank k and degree 1 there is a code α with n_{α} = n, k_{α} = k and some b⃗ ⊧ θ_{α}(y⃗) such that MR(φ_{α}(x⃗, b⃗) △ X) < k_{α}.

Proof. Let X be given. We begin with a formula φ(x⃗, b⃗_{0}) such that b⃗_{0} is a canonical base of the type determined by X and such that MR(φ(x⃗, b⃗_{0}) △ X) < k. Since Morley rank and degree are definable in algebraically closed fields, we may assume that φ satisfies (ii). If, in addition, φ(x⃗, b⃗_{0}) witnesses all algebraic dependencies and equalities between the components of a generic solution, property (iii) holds also. Now, b⃗_{0} is a canonical base if and only if MR(φ(x⃗, b⃗_{0}) △ φ(x⃗, b⃗′)) < k → b⃗_{0} = b⃗′ for every b⃗′ ⊧ tp(b⃗_{0}). Thus φ satisfies (iv), if we add some finite part of q(y⃗) = tp(b⃗_{0}) to φ(x⃗, y⃗).

Choose generic realizations a⃗_{1}, …, a⃗_{m} of φ(x⃗, b⃗_{0}), independent over b⃗_{0}. If m is large enough, we have b⃗_{0} = f(a⃗_{1}, …, a⃗_{m}) for some 0–definable function f. We strengthen φ, so that b⃗ = f(x⃗_{1}, …, x⃗_{m}) for every sequence x⃗_{1}, …, x⃗_{m} of independent generic realizations of φ(x⃗, b⃗), and set φ_{α} = φ.

Finally let ψ_{α}(x⃗_{1}, …, x⃗_{m}, b⃗) express

“For every sequence x⃗_{m+1}, …, x⃗_{2m} of generic realizations of φ_{α}(x⃗, b⃗), such that x⃗_{1}, …, x⃗_{2m} is independent over b⃗, and every choice of distinct indices i_{1}, …, i_{m} ∈ {1, …, 2m}, we have b⃗ = f(x⃗_{i_{1}}, …, x⃗_{i_{m}})”. □

Let α be a code and σ a permutation of {1, …, n_{α}}. We denote by α^{σ} the code obtained from α by permuting each of the tuples x⃗, x⃗_{1},…,x⃗_{m_{α}} in φ_{α} and ψ_{α} according to σ. We call α^{σ} a permutation of α.

Definition 2.4. Two codes α and α′ are equivalent if n_{α} = n_{α′} and m_{α} = m_{α′} and

• for every realization b⃗ of θ_{α} there is a tuple b⃗′ such that (in ℂ) φ_{α}(x⃗, b⃗) ≡ φ_{α′}(x⃗, b⃗′) and ψ_{α}(x⃗_{1}, …, x⃗_{m_{α}}, b⃗) ≡ ψ_{α′}(x⃗_{1}, …, x⃗_{m_{α}}, b⃗′).

• the same replacing the roles of α and α′.

The following lemma is a slightly weaker as the statement of Lemma 2 in [6].

Lemma 2.5. There is a set C of codes such that (viii) For each (non–empty) definable set X of Morley degree 1 there is a code α ∈ C

and some b⃗ such that MR(φ_{α}(x⃗, b⃗) △ X) < k_{α}. (ix) If α, α′ ∈ C, ⊧ θ_{α}(b⃗) and MR(φ_{α}(x⃗, b⃗) △ φ_{α′}(x⃗, b⃗′)) < k_{α}, then α′ = α.^{2}

(x) If α belongs to C, then each permutation of α is equivalent to a code in C.

Proof. We refer to the claim of (viii) as “X can be coded by α”. List all non–empty definable sets of degree 1 up to conjugation by automorphisms of ℂ by X_{1}, X_{2}, … This is possible since ACF_{q} is small, i.e. it has only countably many n-types for each n. It is enough to show that each X_{i} can be coded by some elements of C. We will obtain C as the union of a sequence ∅ = C_{0} ⊂ C_{1} ⊂ ⋯ of finite sets of codes, constructed as follows. Assume that C_{i−1} has been constructed and it is closed under permutations in the weak sense of (x). If X_{i} can be coded by an element of C_{i−1}, we set C_{i} = C_{i−1}. Otherwise, choose a code α and b⃗_{0} such that MR(φ_{α}(x⃗, b⃗_{0}) △ X) < k_{α}. We replace φ_{α} by

2We identify two codes if there defining formulas are equivalent in ℂ.

φ_{α}(x⃗, y⃗) ∧ “{x⃗ | φ_{α}(x⃗, y⃗)} cannot be coded by an element of C_{i−1}”.

and obtain a new code, which still codes X_{i}. We may assume that no permutation of α can code a set which can also be coded by a code in C_{i−1}. Let G be the group of all σ ∈ Sym(n_{α}) such that

MR(φ_{α}(x⃗, b⃗_{0}) △ φ_{α^{σ}}(x⃗, σb⃗_{0})) < k_{α}

for some element denoted as σb⃗_{0} which has the same type as b⃗_{0}. After adding a finite part of the type of b⃗_{0} to φ_{α}(x⃗, y⃗) we may assume that for all realizations b⃗ of θ_{α} and all σ, there exists σb⃗ with MR(φ_{α}(x⃗, b⃗) △ φ_{α^{σ}}(x⃗, σb⃗)) < k_{α} iff σ ∈ G. Note that σb⃗ is a ∅-definable function of b⃗. If we let permutations act on the right on codes, this defines a left action of G on θ_{α}(ℂ).

It is easy to check that

φ_{β}(x⃗, y⃗) = ⋀_{σ∈G} φ_{α^{σ}}(x⃗, σy⃗)

and ψ_{β}(x⃗_{1}, …, y⃗) = ⋀_{σ∈G} ψ_{α^{σ}}(x⃗_{1}, …, σy⃗) defines a code, which again codes X. Also, for σ ∈ G, we have φ_{β}(x⃗, y⃗) ≡ φ_{β^{σ}}(x⃗, σy⃗) and ψ_{β}(x⃗_{1}, …, y⃗) ≡ ψ_{β^{σ}}(x⃗_{1}, …, σy⃗), which shows that β is equivalent to β^{σ}. Now choose representatives ρ_{1}, …, ρ_{r} for the right cosets of G in Sym(n_{α}) and set C_{i} = C_{i−1} ∪ {β^{ρ_{1}}, …, β^{ρ_{r}}}. □

Remark 2.6. Note that the proof holds in a more general setting of a countable strongly minimal theory with the DMP (definable multiplicity property) where imaginary parameters b⃗ are allowed. It is not possible to find C closed under permutations (as stated in [6]).

3. δ-nonsense

Let X be a set. A function δ : P_{fin}(X) → ℤ is a δ-function if it satisfies the following:

(1) δ(∅) = 0

(2) δ(A ∪ B) + δ(A ∩ B) ≤ δ(A) + δ(B)

Moreover, if for all A we have that δ(A) ≥ 0, then we say that δ is nonnegative. For finite subsets A and B, we define the relative δ-value of A over B by:

δ(A/B) = δ(A ∪ B) − δ(B)

Now, (2) is equivalent to δ(A/B) ≤ δ(A/A ∩ B). It is easy to see that for any A ∩ B ⊂ C ⊂ B, we have that δ(A/B) ≤ δ(A/C).

Hence, we can extend the definition of the relative δ to subsets Y (possibly not finite) as follows:

δ(A/Y) = inf_{A∩Y ⊂C⊂Y} δ(A/C)

Note that δ(A/Y) is in {−∞} ∪ ℤ. Using notation from [7], we say that Y is self-sufficient in X (denoted as Y ≤ X) if for all finite A ⊂ X, we have that δ(A/Y) ≥ 0. We have that δ is nonnegative iff ∅ ≤ X.

Y is self-sufficient iff δ(A) ≥ δ(A ∩ Y) for all A. If Y ≤ X, it follows that Y ∩Z ≤ Z for all Z. Hence, self-sufficiency is transitive. Moreover, the intersection of self-sufficient sets is again self-sufficient and each set S is contained in a smallest self-sufficient subset, its self-sufficient closure cl_{X}(S). If δ is nonnegative, finite sets have finite closures.

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

δ(Z/Y′) < 0 for all Y ⊊ Y′ ⊊ Z

.

4. Black points

We extend the ring language L to L^{∗} = L ∪ {N}, where N is a unary predicate. All considered L^{∗}–structures are colored subsets of ℂ, i.e. subsets A of ℂ endowed with an interpretation N(A) for N (les points noirs). The notation A ⊂ B implies N(A) = A ∩ N(B).

We want to amalgamate à la Fraïssé-Hrushovski finite L^{∗}–structures A according to a function δ defined as follows:

δ(A) = p · trdeg A − |N(A)|

Note that δ satisfies conditions (1) and (2) from Section 3. With this particular definition, we have that δ({a}) ≤ p. We are in a setting as in the previous section.

Although the general amalgam was studied in careful detail in [11], we will concentrate on the collapse closer to the spirit of [6]. Hence, we will consider just sets, and not the L^{∗}-substructures that they generate. Nonetheless, in an abuse of notation, we will call them L^{∗}-structures (and not partial L^{∗}-structures).

All the lemmas in the rest of the section are true for arbitrary, finite or infinite, L^{∗}–structures.

Lemma 4.1. Let B ≤ A be a minimal extension. We have one of the following cases:

(1) If A contains a white point a not in B, then A = B ∪ {a}. Moreover, δ(A/B) = 0 or p, depending whether a is algebraic or transcendental over B.

(2) Otherwise, A = B ∪ {a_{1}, …, a_{n}} with a_{1}, …, a_{n} distinct black and 0 ≤ δ(A/B) ≤ p − 1. Moreover, for any ∅ ≠ S ⊊ {a_{1}, …, a_{n}}, we have that

p · trdeg(A/B ∪ S) < n − |S|.

If δ(A/B) = p − 1, then A = B ∪ {a} with a transcendental over B and black.

Proof. Recall that B ≤ A is minimal if it is proper and for any B ⊊ A′ ⊊ A, we have that δ(A/BA′) < 0. Equivalently, δ(A/B) is the minimum among all values of δ(A′/B), where B ⊊ A′ ⊂ A, and it is attained only at A.

If a in A\B is white, case (1) follows, since we have that δ(A\{a}/B) ≤ δ(A/B), hence A = B ∪ {a}. The two possibilities for δ(A/B) are now clear.

Let us assume that A \ B contains no white point. Take some a ∈ A \ B. Since B ≤ A, it follows that a is transcendental over B and δ(a/B) = p − 1. By minimality, δ(A/B) ≤ p − 1.

If δ(A/B) = p − 1, then clearly A = B ∪ {a}.

□

Definition 4.2. A minimal extension B ≤ A of type (2) is good if tp(A/B) is stationary and δ(A/B) = 0. A code α is good if it is “the code” of a good minimal extension. That is,

• n_{α} = pk_{α}.

• φ_{α}(x⃗, y⃗) implies that all x_{i}’s are different and different from the components of y⃗.

• If ⊧ φ_{α}(a⃗, b⃗), for each ∅ ≠ s ⊊ {1, …, n_{α}}, we have

p · trdeg(a⃗/a⃗_{s}b⃗) < (n_{α} − |s|).

Note that, by (iii), the last two conditions are true, if they hold for just one realization b⃗ of θ_{α} and one generic realization a⃗ of φ_{α}(x⃗, b⃗).

Let C_{g} be the subset of good codes in C.

The next lemma is clear from the definitions.

Lemma 4.3. Let α be a good code, b⃗ ∈ dcl(B) realize θ_{α}, and a⃗ be a B–generic black realization of φ_{α}(x⃗, b⃗). Then, B ∪ {a_{1}, …, a_{n_{α}}} is a good extension of B.

Lemma 4.4. Let B ≤ A = B ∪ {a_{1}, …, a_{n}} be a good extension. Then there is a good code α and b⃗ ∈ dcl(B) such that ⊧ φ_{α}(a⃗, b⃗).

Proof. Choose χ(x⃗) ∈ tp(a⃗/B) of Morley rank k = MR(a⃗/B) and degree 1. There is α ∈ C and b⃗ such that MR(χ(x⃗) △ φ_{α}(x⃗, b⃗)) < k = k_{α}. Since a⃗ is a B– generic realization of χ(x⃗), it is also a B–generic realization of φ_{α}(x⃗, b⃗). Since b⃗ is a canonical base of tp(a⃗/B), b⃗ belongs to dcl(B). Since A/B is good, we have that α is a good code. □

In the previous Lemma, we chose a⃗ as a B–generic realization of φ_{α}(x⃗, b⃗). The following result shows that this the only possibility.

Lemma 4.5 (cf. Lemma 3A in [6]). Let α be a good code, b⃗ ∈ acl(B) realize θ_{α}, and a⃗ be a black 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, then a⃗ ∩ B = ∅ and a⃗ is a B–generic realization of φ_{α}(x⃗, b⃗).

Proof. If a⃗ is not disjoint from B we have

δ(a⃗/B) ≤ δ(a⃗/a′b⃗) < 0.

for a⃗ ∩ B = a′. Hence, δ(a⃗/B) = 0 yields that a⃗ ∩ B = ∅. In this case, we have δ(a⃗/B) ≤ p·k_{α} −n_{α} = 0. Therefore, trdeg(a⃗/B) = k_{α} and a⃗ is a B–generic solution of φ_{α}(x⃗, b⃗). □

5. The (in)famous μ function

We now fix a function μ^{∗} : C_{g} → ℕ which is finite-to-one on the set of all α with n_{α} = n for each n in ℕ. Moreover, μ^{∗}(β) = μ^{∗}(α) must hold if β is equivalent to a permutation of α, and

μ^{∗}(α) ≥ m_{α} − 1.

The function μ is then defined by

μ(α) = ((p − 1)(n_{α} − 1) + 1)m_{α} + μ^{∗}(α).

We note that μ(α) ≥ m_{α}.

Note 5.1. One can replace in the following μ by μ′(α) = F (α) + μ^{∗}(α) for any function F which satisfies F (α^{σ}) = F (α) and F (α) ≥ ((p − 1)(n_{α} − 1) + 1)m_{α}. The class of functions μ is not increased by this, only the complete theories T^{μ} (see Section 7) get weaker, but equivalent, axiomatizations.

We recover the definition introduced in [11] for approximations to a Morley sequence of a given good minimal extension.

Definition 5.2. Let α be a good code and b⃗ ⊧ θ_{α}. A pseudo-morley sequence for α over b⃗ is a (finite) sequence a⃗_{1}, …, a⃗_{r} of disjoint realizations of φ_{α}(x⃗, b⃗) painted in black such that any distinct m_{α} elements among {a⃗_{1}, …, a⃗_{r}} realize ψ_{α}(x⃗_{1}, …, x⃗_{m_{α}}, b⃗).

It follows that b⃗ is in the definable closure of the pseudo-morley sequence if r ≥ m_{α} from part (v) of 2.1.

We now consider the class of L^{∗}-structures on which δ is non-negative and for any good code in C, we cannot find a pseudo-morley sequence that is longer than the value of μ at this code.

Definition 5.3. The class K^{μ} is the class of all L^{∗}-structures M (i.e colored subsets of ℂ) such that:

• ∅ ≤ M.

• No α in C_{g} has a pseudo-morley sequence in M of length longer than μ(α).

We denote by K_{fin}^{μ} the class of all finite L^{∗}-structures in K^{μ}.

Recall that the first condition means that for any finite set A ⊂ M, we have δ(A) ≥ 0. Clearly, K_{fin}^{μ} is not empty (∅ is an element of this class). In fact all finite subsets of ℂ with no black points are in the class.

Since ACF_{q} is small, K_{fin}^{μ} contains at most countably many structures up to isomorphism.

The following result resumes the ingredients used in [6] stating them in a form closer to the original idea of Fraïssé’s amalgamation procedure to construct a countable ultrahomogeneous model whose age is exactly K_{fin}^{μ}. Moreover, it yields explicit conditions for an L^{∗}-structure to be a member of K^{μ}, which will be useful for exhibiting an axiomatization of this class.

Lemma 5.4. Let M be in K^{μ} and M ≤ M′ a minimal extension.

If M′ contains a new white point, then M′ is in K^{μ}.

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

a) There is a code α ∈ C_{g} and a realization b⃗ ∈ dcl(M) of θ_{α}, such that: i) M′ \ M contains a realization a⃗ of φ_{α}(x⃗, b⃗). ii) M contains a pseudo-morley sequence for α over b⃗ of length μ(α).

b) There is some code α ∈ C_{g} and a pseudo-morley sequence for α in M′ of length μ(α) + 1, such that there are more than μ^{∗}(α) many elements of the sequence contained in M′ \ M.

If a) holds, a⃗ is an enumeration of M′\M and an M–generic realization of φ_{α}(x⃗, b⃗).

Since n_{α} ≥ p for good codes, the lemma implies that M′ ∈ K^{μ} if δ(M′/M) = p − 1.

Proof. If M′ contains a new white point a, by 4.1 (1), we get that M′ = M ∪ {a}. If M′ is not in K^{μ}, it contains a pseudo-morley sequence of length μ(α) + 1 for some code α ∈ C_{g}. Since {a} adds no black points, the sequence is contained in M, which is a contradiction.

Suppose now that M′ \ M has no new white points. If b) holds, M′ is not in K^{μ} by definition. If we have case a), a⃗ is generic over M by Lemma 4.5 (2) and we can extend the sequence of ii) by a⃗, thanks to condition (vii). This shows that M′ is not in K^{μ}. Also, since M′/M is minimal and δ(a⃗/M) = 0, we have that M′ = M ∪ {a_{1}, …, a_{n_{α}}}.

For the other direction, if M′ as above is not in K^{μ}, there exists a code α ∈ C_{g} and a pseudo-morley sequence e⃗_{0}, …, e⃗_{μ(α)} for α in M′ over some b⃗ ∈ dcl(M′). We may rearrange the sequence as follows:

• e⃗_{0}, …, e⃗_{r_{0}−1} are contained in M.

• e⃗_{r_{0}}, …, e⃗_{r_{1}−1} are not in M, but have at least one coordinate in M.
• e⃗_{r_{1}}, …, e⃗_{μ(α)} are in {a_{1}, …, a_{n_{α}}}.

Since M is in K^{μ}, we have that r_{0} ≤ μ(α). There are two possibilities:

Case 1. m_{α} ≤ r_{0}. In this case, b⃗ ∈ dcl(e⃗_{0}, …, e⃗_{m_{α}−1}) is in dcl(M). By 4.5 (1), we have that δ(e⃗_{r_{0}}/M) ≤ 0. Since M ≤ M′, we have that δ(e⃗_{r_{0}}/M) = 0. Hence, for each i, we conclude from 4.5 (2) that either e⃗_{i} is disjoint from or contained in M. That is, r_{0} = r_{1}. As above, we conclude M′ = M ∪ e⃗_{r_{0}}. Hence, r_{0} = μ(α) by disjointness of the pseudo-morley sequence. Therefore, a) holds.

Case 2. r_{0} ≤ m_{α}. Define δ(i) = δ(e⃗_{i}/Me⃗_{0}, …, e⃗_{i−1}). Then, since M ≤ M′, we have that:

0 ≤ δ(e⃗_{0}, …, e⃗_{r_{1}−1}/M) = ∑_{i<r_{1}} δ(i) = ∑_{i<m_{α}} δ(i) + ∑_{m_{α}≤i<r_{1}} δ(i)

For i < m_{α}, we have that δ(i) ≤ (p − 1)(n_{α} − 1) (Note that if d⃗ is a tuple of black points, we always have that δ(d⃗/B) ≤ (p − 1) · trdeg(d⃗/B) for any set B).

For m_{α} ≤ i < r_{1}, it follows that b⃗ belongs to dcl(Me⃗_{0} … e⃗_{i−1}). But there is some coordinate of e⃗_{i} in M, and hence, again from Lemma 4.5 (2), we conclude that δ(i) < 0.

From the inequalities above, we get:

0 ≤ (p − 1)(n_{α} − 1)m_{α} − (r_{1} − m_{α})

That is, r_{1} ≤ ((p − 1)(n_{α} − 1) + 1)m_{α}. Now,

μ(α) − r_{1} + 1 ≥ μ(α) − ((p − 1)(n_{α} − 1) + 1)m_{α} + 1 ≥ μ^{∗}(α) + 1

This yields b). □

Corollary 5.5. Let M be in K^{μ}, α ∈ C_{g}, b⃗ ∈ dcl M a realization of θ_{α} and a⃗ a black M–generic realization of φ_{α}(x⃗, b⃗). Then M′ = M ∪ {a_{1}, …, a_{n_{α}}} is in K^{μ} if and only if none of the following two conditions holds:

a) M contains a pseudo-morley sequence for α over b⃗ of length μ(α). b) There is some code β ∈ C_{g} and a pseudo-morley sequence for β in M′ of length μ(β) + 1, such that there are more than μ^{∗}(β) many elements of the sequence contained in M′ \ M.

Proof. M′ is a minimal extension of M by 4.3, so we can apply the last lemma. We need only show the following: If α′ is a code in C_{g}, b⃗′ ∈ dcl(M) such that a⃗ is a permuted M-generic realization of φ_{α′}(x⃗, b⃗′) and if α′ has a pseudo-morley sequence of length μ(α′) in M over b⃗′, then α has a pseudo-morley sequence of length μ(α) in M over b⃗.

Let σ be a permutation of α such that a⃗ realizes φ_{α^{σ}}. By (x) there is a code α′′ ∈ C_{g} which is equivalent to α^{σ}. So there is b⃗′′ such that φ_{α^{σ}}(x⃗, b⃗′) ≡ φ_{α′′}(x⃗, b⃗′′) and ψ_{α^{σ}}(x⃗_{1}, …, b⃗′) ≡ ψ_{α′′}(x⃗_{1}, …, b⃗′′). The permuted pseudo-morley sequence of α′ is a pseudo-morley sequence of α′′ over b⃗′′ ∈ dcl(M), and a⃗ is an M-generic realization of φ_{α′′}(x⃗, b⃗′′). The properties (ix) and (iv) of C imply α′′ = α and b⃗′′ = b⃗. Finally, we have μ(α) = μ(α′′) = μ(α^{σ}). □

6. Fraïssé limits for K^{μ}

In this section, we show that the class K^{μ} (and hence, K_{fin}^{μ}) has the Amalgamation Property, and hence, we can obtain rich fields as introduced by Poizat in [11] (We apologize for translating notation into other languages).

An isomorphism between two colored subsets A and B of ℂ is a bijection which maps N(A) onto N(B) and is elementary as a partial map defined on ℂ. A selfsufficient embedding from A to B is an isomorphism between A and a self-sufficient subset of B.

Theorem 6.1. 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. By splitting the extension B ≤ A into minimal ones, we may assume it is minimal.

Case 1. B ≤ A has a new white point a. Let p be the type of a over B. We distinguish two (non-exclusive) cases.

Subcase 1.1. p is algebraic and realized in M, say by a′. Self-sufficiency of B in M yields that a′ is white. So A′ = B ∪ {a′} is isomorphic to A. Since δ(a′/B) = 0, it implies that A′ ≤ M.

Subcase 1.2. p can be realized in an extension of M by a new element a′. We paint a′ white and set M′ = B ∪ {a′}.

Case 2. B ≤ A has no new white points. Since B is self-sufficient in A, no element of A \ B is algebraic over B. So we can take for M′ be the free amalgam (as in [11]) of M and A over B, that is, we assume M and A to be algebraically independent over B and let M′ be their union. It is easy to see that M and A are self-sufficient in M′ and that M′/M is minimal. We are done if M′ belongs to K^{μ}. Otherwise, by Lemma 5.4, there are two cases:

Subcase 2.1) There is a code α ∈ C_{g}, a realization b⃗ of θ_{α} in dcl(M), a pseudomorley sequence for α in M over b⃗ of length μ(α) and M′ \ M = a⃗ is a M–generic realization of φ_{α}(x⃗, b⃗). Since a⃗ is independent from M over B and b⃗ is the canonical parameter of tp(a⃗/M)), we have that b⃗ ∈ acl(B) . The sequence cannot be contained in B, since A is in K^{μ}. Hence, there is some black realization a′ of φ_{α}(x⃗, b⃗) in M not completely contained in B. Now, δ(a′/B) = 0 since B ≤ M, therefore a′ is generic over B by Lemma 4.5. So, B ∪ {a′_{1}, …, a′_{n_{α}}} is self-sufficient in M and it is isomorphic to A over B.

Subcase 2.2) There is a code α ∈ C_{g}, a canonical basis b⃗ in dcl(M′) for α such that there is a pseudo-morley sequence e⃗_{0}, …, e⃗_{μ(α)} for α over b⃗ in M′ with more than μ^{∗}(α) many elements coming from M′ \ M. Again, since μ^{∗}(α) + 1 ≥ m_{α}, we have that b⃗ is in dcl(A). There must be at least one member e⃗_{i} not contained in A (because A is in K^{μ}). Since A ≤ M′, it follows from 4.5 (2) that e⃗_{i} is an A–generic realization of φ_{α}(x⃗, b⃗) in M. But e⃗_{i} and A are independent over B, therefore the canonical basis b⃗ of α is in acl(B). Pick some e⃗_{j} in M′ \ M = A \ B. Again it follows that e⃗_{j} is a B–generic realization of φ_{α}(x⃗, b⃗). Since M′/M is minimal, e⃗_{j} enumerates M′ \ M and we are in subcase 2.1. Note that all e_{k}, k ≠ j, are in M, which implies b⃗ ∈ dcl(M). □

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 6.2. There is a unique (up to isomorphism) countable rich structure M in K^{μ}.

We will see in Theorem 7.2 that rich structures are colored algebraically closed fields. We will call them rich fields.

Remark 6.3. Let M be a rich field^{3}, α be a code in C_{g} and b⃗ be a realization of θ_{α} in M. Let dim_{α}(M/b⃗) be the maximal length of a pseudo-morley sequence of α over b⃗ in M and B = cl_{M}(b⃗) the (finite) self-sufficient closure of b⃗ in M. Then there are two cases, either

dim_{α}(M/b⃗) = dim_{α}(B/b⃗)

or

dim_{α}(M/b⃗) = μ(α).

Proof. Choose a black B–generic realization a⃗ of φ_{α}(x⃗, b⃗) outside M. If A = B ∪ a⃗ is not in K^{μ}, all realizations of φ_{α}(x⃗, b⃗) in M are contained in B (by Lemma 4.5 (2)). Therefore dim_{α}(M/b⃗) = dim_{α}(B/b⃗).

If A belongs to K^{μ}, let C ≤ M be a finite extension of B with dim_{α}(M/b⃗) = dim_{α}(C/b⃗) and let C′ be the free amalgam of C and A over B. Since M is rich, C′ does not belong to K^{μ}. The proof of 6.1 (applied to C instead of M) and of 5.5 shows that dim_{α}(C/b⃗) = μ(α). □

3The remark is true for all models of T^{μ}, as defined in Section 7.

7. A theory for K^{μ}

In this section, we will show that the class K^{μ} is axiomatizable and we will give explicit axioms that describe some completion. Rich fields will then be ω-saturated models of this theory. First, a foreword about the choice of axioms:

We will see in Section 8 that extensions with δ = 0 will become algebraic. We know (by reducing it to the case of good minimal extensions) that at most there are μ many realizations. If we are given a minimal extension B ≤ A, where B ≤ M, we could amalgamate A and M freely over B and the amalgam could be potentially an element of K^{μ}. By richness, this cannot happen, since there is one realization too many in the amalgam not in M. Hence, we need to prohibit the amalgam to be an element of K^{μ}. We know exactly by 5.4 and 5.5 when this happens. Therefore, our axioms should state that such an amalgam cannot happen.

The theory T^{μ} in the extended language L^{∗} = L ∪ {N} has the following axioms (more precisely, axiom schemes):

Universal Axioms: (1) Any model is an integral domain of characteristic q.

(2) ∅ is self-sufficient in any model of T^{μ}.

(3) Given a code α ∈ C_{g}, any pseudo-morley sequence for α has length at

most μ(α). ∀∃ Axioms:

(4) Any model of T^{μ} is an algebraically closed field of characteristic q.

(5) Given a code α ∈ C_{g} and b⃗ realizing θ_{α}(y⃗), one of the following holds: a) α has a pseudo-morley sequence of length μ(α) over b⃗.

b) Given a realization a⃗ of φ_{α}(x⃗, b⃗) generic over the model that we are considering, if we paint a⃗ in black, there is a code β ∈ C_{g} and a pseudo-morley sequence for β of length μ(β) + 1 in the L^{∗}-structure consisting of the model and a⃗ such that there are more than μ^{∗}(β) many elements of the sequence contained {a_{1}, …, a_{n_{α}}}.

Note 7.1. We discuss here why the above axioms are first-order and their meaning. Since our final theory will have finite Morley rank, it follows from [10] that Axiom (4) needs to be included. Axiom (3) will yield that the types of δ = 0 will become algebraic, and hence of Morley rank 0.

Why is Axiom (5) axiomatizable? In order to encode β, we need to determine a priori how many variables we will use. Equivalently, how many β’s need to be considered. We cannot use more than n_{α} variables. On the other hand, we have n_{β} many variables to consider for each element of the pseudo-morley sequence, and there are at least μ^{∗}(β) + 1 many such members. That is,

(μ^{∗}(β) + 1)n_{β} ≤ n_{α}

By the finite-to-one condition on μ^{∗}, there are only finitely many β’s that satisfy the above inequality, and we are done.

Moreover, it follows from 6.3 that in order to get a complete theory, we do not need to determine how many realizations of a code there must be in a model, since we implicitly do so.

Theorem 7.2. An L^{∗}-structure is rich if and only if it is an ω-saturated model of T^{μ}.

Proof. Let M ⊧ T^{μ} be ω-saturated. Let B ≤ M and B ≤ A be finite sets. We need to find a self-sufficient B-copy of A in M. Splitting B ≤ A into minimal extensions, we are reduced to the minimal case. We can distinguish four different cases: If B ≤ A is algebraic, we are done (by Axiom (4)). If B ≤ A = B ∪ {a⃗} is of type (2) (see Lemma 4.1) with δ(A/B) = 0, consider the free amalgam M′ of M and A over B. Since M is algebraically closed, M′ is a good extension of M. By (the proof of) Lemma 4.4 and 4.5 there is a code α ∈ K_{g} and b⃗ ∈ M such that a⃗ is an M-generic realization of φ_{α}(x⃗, b⃗). By Axiom (5) M′ does not belong to K^{μ}. Theorem 6.1 implies that A has a strong embedding over B into M. For 1 ≤ δ(A/B) ≤ p − 1, we need to approximate the extension by extensions of δ < δ(A/B) and apply induction. We know by 4.1 that A contains no new white points. Choose some element a ∈ A \ B. Since a is transcendental over B, a^{n} is not in A for large n. We can paint a^{n} in black and consider A_{n} = A ∪ {a^{n}}. It is easy to check that B ≤ A_{n} is minimal and δ(A_{n}/B) < δ(A/B). The sequence A_{n}/B converges (in the space of L-types) to the extension A_{∞}/B, where A_{∞} = A ∪ {c}, with c transcendental over A and black. Clearly A ≤ A_{∞} ∈ K^{μ}, by 5.4. Since there is only a finite number of codes α ∈ C_{g} for which there could be a pseudo-morley sequence of length longer than μ(α) in any A_{n} (bounded only in terms of |A|), we have that A_{n} is in K^{μ} for large n. Hence, by induction, we can find self-sufficient B-copies of A_{n} in M for large n. By saturation of M, A_{∞} is also self-sufficiently embedable over B. Since A ≤ A_{∞}, we conclude that there is a self-sufficient B-copy of A in M. For the last case, let A = B ∪ {a} with a white transcendental over B. Consider for each n the extension

B ≤ B ∪ {c} ≤ B ∪ {c, c^{n}},

where c is black transcendental over B and c^{n} is white. B ∪{c, c^{n}} belongs to K^{μ} by 5.4. By the above, that we can realize B ≤ B ∪ {c, c^{n}} self-sufficiently in M. Since these extensions converge to B ≤ B ∪ {c, a} = A′ where c and a are algebraically independent over B, we can realize B ≤ A′ self-sufficiently in M. Since A ≤ 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 closure of B in M, we can assume that B ≤ M. Paint a 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 (4). For Axiom 5, let α and b⃗ be as in the statement such that neither a) nor b) hold. Choose some generic black realization of φ_{α}(x⃗, b⃗) over M. By 5.5, we have that M ∪ a⃗ is in K^{μ}. Choose some finite 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 pseudomorley sequence in M for α of arbitrarily large length. This contradicts that M is in K^{μ}. Now, M is elementarily equivalent to an ω–saturated structure M′, which is by the above a model of T^{μ} and therefore rich. So M is ∞-equivalent to M′ and therefore ω–saturated itself. □

Corollary 7.3. Let M be an L^{∗}-structure in K^{μ}. Then M ⊧ T^{μ} iff every existential L^{∗}(M)-formula ϕ true in some M ≤ N with N ⊧ “Universal Axioms of T^{μ}” holds also in M.

Proof. Let M be a model of T^{μ}, N and ϕ as above. We can assume M and N are saturated. Let B ≤ M contain all parameters in ϕ. Choose some B ⊂ A ≤ N containing a realization of ϕ. Since B ≤ A and M is rich, we can embed A in M over B. Hence, we have a solution for ϕ in M.

If M is existentially closed among self-sufficient extensions, it satisfies Axiom (4), since acl(M) (new elements painted in white) is a self-sufficient extension. If M does not satisfy Axiom (5), there is an α ∈ C_{g}, b⃗ ∈ M and a black M-generic solution a⃗ of φ_{α}(x⃗, b⃗) such that M ≤ M ∪a⃗ is in K^{μ}. Considering finite sets C ≤ M containing b⃗ and using the existential closedness of M, we can find infinitely many disjoint black realizations of φ_{α}(x⃗, b⃗). (We may assume that φ_{α} is quantifier free.) This contradicts that M ∈ K^{μ}. □

Corollary 7.4. The theory T^{μ} is complete.

Two tuples a⃗ and b⃗ in two models M and N of T^{μ} have the same L^{∗}-type iff there is an isomorphism f : cl(a⃗) → cl(b⃗) which maps a⃗ to b⃗.

An extension M ⊂ N of models of T^{μ} is elementary iff M is self-sufficient in N^{4}

Proof. T^{μ} is complete, since any two countable saturated models are elementarily equivalent, by richness.

Consider two models M, N of T^{μ}. If M is subset of N, but no self-sufficient, there is a finite A ≤ M and a tuple b⃗ ∈ N sucht that δ(b⃗/A) < 0. Responsible is a finite part of the L^{∗}-type of b⃗ over A. So M ≺ N would imply the existence of an a⃗ ∈ M with δ(a⃗/A) < 0, which is not possible.

If a⃗ and b⃗ have the same L^{∗}–type, it is easy to see that the map a⃗ ↦ b⃗ extends to an isomorphism f : cl(a⃗) → cl(b⃗). Conversely, let f be given. Choose rich extensions M ≺ M′ and N ≺ N′. We know that M and N are self-sufficient in these extension and therefore also cl(a⃗) and cl(b⃗). Since isomorphisms between finite self–sufficent subsets of M′ and N′ have the back-and-forth property, f is elementary map.

Finally assume that M ≤ N. Since cl_{M} is the restriction of cl_{N} to M, all finite tuples a⃗ ∈ M have the same L^{∗}-type in M as in N, i.e. M ≺ N. □

8. Computing ranks

In this section, we compute the Morley rank of types in T^{μ}. In order to avoid confusion, we will denote it by MR^{∗}, since we work with L^{∗}-types tp^{∗}(a⃗/B). We work inside a sufficiently saturated model M of T^{μ}.

Lemma 8.1. T^{μ} has finite Morley rank.

Proof. It is clear that cl(A) is contained in acl^{∗}(A). This implies

MR^{∗}(a⃗/C) = MR^{∗}(cl(Ca⃗)/ cl(C)).

4By an observation of M. Hils T^{μ} is model complete, i.e. all extensions of models of T^{μ} are self-sufficient. See Remark 8.4 for a proof.

So it is enough to compute Morley ranks MR^{∗}(A/B), where B ≤ A ≤ M and A \ B is finite. We will show that the rank is bounded by a function of δ(A/B).

We prove first that δ(A/B) = 0 implies that A is algebraic in M over B, i.e. MR^{∗}(A/B) = 0. For this we may assume that A/B is minimal. If A has a new white element, then A/B is algebraic (in the field sense). Otherwise, A\B contains only black points and we may assume – after adding algebraic elements to B – that A/B is good. By 4.4, A \ B is enumerated by a generic solution a⃗ of a code α ∈ C_{g} over some b ∈ dcl(B). By 4.5 any sequence A_{i} of different conjugates of A over B in M yields a sequence of B–generic realizations of φ_{α}(x⃗, b⃗) which is (in ℂ) independent over B. So the sequence is a pseudo-morley sequence of α over b⃗ and cannot be longer that μ(α). This proves that A/B is algebraic in M.^{5}

Now, assume that δ(A/B) = d > 0 and that MR^{∗}(A′/B′) ≤ r for all B′ ≤ A′ ≤ M with δ(A′/B′) < d. The above case shows that we may also assume that for all C ≤ A with B ⊊ C ⊊ A we have δ(C/B) > 0 and δ(A/C) > 0. We distinguish three cases:

Case 1: A/B is not minimal. Then we find B ≤ C ≤ A with δ(C/B) < d and δ(A/C) < d. Enumerate C \ B by c⃗ and A \ C by a⃗ and choose L^{∗}-formulas ρ(z⃗) and χ(z⃗, x⃗) over B which are satisfied by c⃗ and c⃗a⃗ and which imply δ(z⃗/B) < d and δ(x⃗/Bz⃗) < d. By above inductive assumption, we have MR^{∗}(c′/B) ≤ r and MR^{∗}(a′/Bc′) ≤ r for all realization c′a′ of ρ(z⃗) ∧ χ(z⃗, x⃗). Now we can apply Erimbetov’s inequalities [5] and obtain MR^{∗}(ρ(z⃗) ∧ χ(z⃗, x⃗)) ≤ r(r + 1). Hence MR^{∗}(A/B) ≤ r(r + 1).

Case 2: A/B is minimal and 1 ≤ d ≤ p − 1. We fix an enumeration a⃗ of A \ B. We may again assume that p = tp(a⃗/B) is stationary. Choose an L-formula ϕ(x⃗) in p of the same Morley rank and of degree 1 which satifies 2.1 (iii). It follows from Lemma 4.5 that for every black realization a′ of φ, only two possibilities may occur: Either we have δ(a′/B) < d, which implies δ(cl(Ba′)/B) < d and MR^{∗}(a′/B) = MR^{∗}(cl(a′)/B) ≤ r, or we have δ(a′/B) = d, which implies tp(a′/B) = p. If Ba′ is not self-sufficient in M, we conlude again MR^{∗}(cl(a′)/B) ≤ r. Otherwise, by 7.4, we have that tp^{∗}(a′/B) = tp^{∗}(a⃗/B). This shows that the L^{∗}–type of a⃗ over B is not an accumulation point of types over B of rank bigger than r. If B were an ω-saturated elementary substructure of M, we could conclude that MR^{∗}(A/B) ≤ r + 1. Hence, consider any ω-saturated elementary substructure N which contains B. Since A is either contained in N or intersects N in B, we have δ(A/N) ≤ δ(A/B), which implies MR^{∗}(A/N) ≤ r + 1. Since N was arbitrary, it follows MR^{∗}(A/B) ≤ r + 1.

Case 3: A/B is minimal and d = p. Then A = B ∪ {a} with a white and transcendental. All 1-Types different from tp^{∗}(a/B) have rank ≤ r. So, if B were an ω–saturated elementary model, we could conclude MR^{∗}(a/B) ≤ r +1. By the same argument as above, we show that the claim holds. □

For any set of parameters C and any finite tuple a⃗ we define

d(a⃗/C) = δ(cl(Ca⃗)/ cl(C)).

5Note that the number of conjugates of a⃗ over B is bounded by n_{α}! · μ(α).

Theorem 8.2. MR^{∗}(a⃗/C) = d(a⃗/C).

Proof. Since M is a field of finite Morley rank, Morley rank satisfies the Lascar inequalities by a result of Lascar [9]. It follows now from the proof of 8.1 and the additive character of δ that MR^{∗} ≤ d.

For the other inequality, let B ≤ A ≤ M be minimal, and δ(A/B) = d > 0. It follows from the proof of 7.2 (and from 7.4) there is a sequence of extensions B ≤ A_{n} ≤ M, such that δ(A_{n}/B) = d − 1 and

lim_{n→∞} tp^{∗}(A_{n}/B) = tp^{∗}(A/B).

Since MR^{∗}(A_{n}/B) ≥ d − 1 by induction, we have MR^{∗}(A/B) ≥ d. □

Proof of the Main Theorem: For any a in M, we have:

d(a) ≤ δ(a) = \begin{cases}p & if a is white \\ p − 1 & if a is black \end{cases}

This shows MR^{∗}(T^{μ}) ≤ p and MR^{∗}(N) ≤ p − 1. On the other hand the structures {a}, {b} with a white, transcendental and with b black transcendental are both in K^{μ}. So we find them as self-sufficient subsets of M. Then d(a) = δ(a) = p and d(b) = δ(b) = p − 1. This proves the result. □

Example 8.3. It was observed in Theorem 18 [2] that every generic white point is the sum of p independent black points of Morley rank 1. We want to give a simpler proof of this fact.

Let a_{1}, …, a_{p} be generic independent elements of ℂ. It follows trivially from

5.4 that the black A_{i} = {a_{i}^{r}}_{1≤r≤p−1} and ⋃_{i=1}^{p} A_{i} with all elements painted in black

belong to K^{μ}.

Take now a = a_{1} + ⋯ + a_{p} painted in white. Again by 5.4, ⋃_{i=1}^{p} A_{i} ∪ {a} belongs

to K^{μ}. We may assume that ⋃_{i=1}^{p} A_{i} ∪ {a} ≤ M.

Since {a} ≤ M, we have d(a) = p. Hence, a is an generic white element. Since cl(a_{i}) = A_{i} ≤ M, we have d(a_{i}) = δ(A_{i}) = 1. So each a_{i} has Morley rank 1.

Remark 8.4 (added January 23, 2005). Martin Hils made the following oberservation: A theory of fields of finite Morley rank is ℵ_{1}–categorical. Since T^{μ} is ∀∃–axiomatizable, a theorem of Lindström implies that T^{μ} is model complete.

Let us give a direct proof. Assume that M is a model of T^{μ} and N an extension which belongs to K^{μ}. We want to show that M is self–sufficient in N. We may assume that δ(N/M) < 0 and that N is minimal with this property. Then N = N′ ∪{a}, where M ≤ N′ with δ(N′/M) = 0, and a is black and algebraic over N′ in the field sense. Choose a rich field N′ ≤ N′′. Then N′′ is an elementary extension of M. On the other hand, we have MR^{∗}(N′/M) = d(N′/M) = δ(N′/M) = 0, so N′ is, in N′′, algebraic over M and therefore contained in M. This implies that a is field-algebraic over M, which is impossible, since M is an algebraically closed field.

References

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[3] A. Baudisch, M. Ziegler, Hrushovskis Fusion, preprint (2004).

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[5] M. Erimbetov, Complete theories with 1-cardinal formulas, Algebra i Logika, 14, n^{o} 3, 245– 257 (1975).

[6] E. Hrushovski, Strongly minimal expansions of algebraically closed fields, Israel J. Math, 79, 129–151, (1992).

[7] E. Hrushovski, A new strongly minimal set, Annals of Pure and Applied Logic, 62, 147–166, (1993).

[8] E. Hrushovski, B. Zil’ber, Zariski Geometries, Bull. Amer. Math. Soc. 28, 315–323, (1993).

[9] D. Lascar, Les groupes ω-stables de rang fini, Trans. Amer. Math. Soc., 292, 451–462, (1985).

[10] A. Macintyre, On ω_{1}-categorical fields, Fund. Math.,71, 1-25 (1971).

[11] B. Poizat, Le carré de l’égalité , J. Symb. Logic, 64, n^{o} 3, 1338–1355, (1999).

[12] A. Baudisch, A. Martin-Pizarro, M. Ziegler, Hrushovskis Fusion, preprint (2004).

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

  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