Perfect pseudo-algebraically closed fields are algebraically bounded Zoé Chatzidakis – CNRS - Paris 7 Ehud Hrushovski^{1} – Hebrew University

Introduction

The concept of algebraic boundedness of a theory of fields is introduced by Van den Dries in [D]. It is a strong uniformity condition on finite definable sets, see §1 for the definition. If a theory T is algebraically bounded, one can define a good notion of dimension on definable sets of any model K of T: if S ⊆ K^{n} is definable, set dim(S) to equal the algebraic dimension of the Zariski closure S̃ of S. This notion of dimension is then definable in definable families, preserved by definable bijection, and has the main properties expected of a dimension, such as monotonicity and additivity. In particular, the existence of such a dimension precludes the existence of a definable injection K^{2} → K. It follows that a field K whose theory is algebraically bounded is necessarily perfect (if char(K) = p > 0 and a ∈ K \ K^{p}, then (x, y) ↦ x^{p} + ay^{p} is injective).

Classical examples of algebraically bounded theories are the theories of algebraically closed fields, of real closed fields, of the p-adic fields. For more details, see [D].

Recall that a field K is pseudo-algebraically closed (PAC) if every (absolutely irreducible) variety V defined over K has an K-rational point. The model theory of these fields is quite interesting, and has been studied extensively by various authors after their introduction by Ax. In [CDM], Cherlin, Van den Dries and Macintyre determine the elementary theory and the structure of definable sets of PAC fields in terms of the structure of the finite Galois extensions of the field. Note on the other hand that an arbitrary graph can be interpreted in a PAC field via the Galois structure, cf. [CDM] (or see chapter 22 of [FJ]).

In this paper we show that the theory of perfect PAC fields is algebraically bounded. This extends a result of Jarden on perfect Frobenius fields (Theorem (2.6) in [J], see also (5.7) in [ChDM] where it is proved for pseudo-finite fields). The result of this paper was obtained in 1993 in the course of a conversation between the two authors. The first author then lectured on it at Wesleyan University, and benefited from comments from M. Messmer, A. Pillay, P. Scowcroft and C. Wood.

1. Preliminaries In this section, we set up notation, define algebraic boundedness, and briefly recall the

notion of the (complete) inverse system of the absolute Galois group of a field, the logic attached to it, and the results of [CDM] on PAC fields. All the results on complete systems and PAC fields were announced and stated in [CDM], proved in the unpublished preprints [CDM1] and [CDM2]. For a sketch of the proofs, one can however consult Chapter 5 of [C]. We assume familiarity with basic field theory (regular extensions, varieties, etc.), see e.g. Chapter III of Lang’s book [L], and with the very basic notions of model theory (languages, formulas, theories), see any introductory textbook in model theory, such as e.g. [CK] or [M].

1 Supported by the Israel Science Foundation (1.1) Notation. If K is a field, we denote by K^{s} the separable closure of K, by K^{alg} the algebraic closure of K, and by G(K) the absolute Galois group of K, i.e., G(K) = Gal(K^{s}/K). We let L denote the language of rings {+, −, ·, 0, 1}.

(1.2) Algebraic boundedness. A theory T in a language containing the language of rings L, and which contains the theory of fields, is algebraically bounded if, given any formula φ(x̄, y), there are polynomials f_{1}(X̄, Y), …, f_{n}(X̄, Y) ∈ ℤ[X̄, Y] such that, whenever K is a model of T and ā is a tuple of elements of K such that φ(ā, K) =_{def} {y ∈ K | φ(ā, y)} is finite, then there is an index i such that the polynomial f_{i}(ā, Y) is not identically 0 on K and φ(ā, K) is contained in the set of roots of f_{i}(ā, Y) = 0.

(1.3) The inverse system of a profinite group. We consider an ω-sorted language L_{G} = {≤, P, C, 1}, with sorts indexed by the positive integers, and where ≤, C are binary relations, P is a ternary relation, and 1 is a constant symbol of sort 1.

Recall that a profinite group is a topological group which is compact, Hausdorff and totally disconnected; equivalently it is a topological group which is the inverse limit of finite groups; see e.g. [RZ] for definitions and further properties.

Let G be a profinite group. We consider the set S(G) = ⋃ · G/N, where N ranges over the set N of open normal subgroups of G. If N ⊆ M are in N, denote by π_{N,M} the natural epimorphism G/N → G/M. We denote the elements of G/N by gN, g ∈ G.

We define an L_{G}-structure on S(G) as follows:

— gN is of sort n if and only if [G : N] ≤ n. Note that we do not assume that the sorts are disjoint.

— gN ≤ hM if and only if N ⊆ M.

— C(gN, hM) if and only if N ⊆ M and gM = hM.

— P (g_{1}N_{1}, g_{2}N_{2}, g_{3}N_{3}) if and only if N_{1} = N_{2} = N_{3} and g_{1}g_{2}N_{1} = g_{3}N_{1}.

— 1 = G.

Note that the L_{G}-structure S(G) encodes precisely the inverse system {G/N, π_{N,M} | N, M ∈ N , N ⊆ M}, and therefore determines G uniquely, since G = lim_{←}G/N. We call the L_{G}-structure S(G) the complete system associated to G. The functor G ↦ S(G) defines a duality between the category of profinite groups with continuous epimorphisms, and a certain elementary class of L_{G}-structures, with L_{G}-inclusions as morphisms.

(1.4) The complete system of the absolute Galois group of a field. Let K be a field. The complete system S(G(K)) is then simply ⋃ · Gal(L/K), where L ranges over all finite Galois extensions of K. The group epimorphisms encoded by the binary relation C correspond to the restriction maps Gal(M/K) → Gal(L/K) whenever M is a finite Galois extension of K containing L. The largest element of S(G(K)) is Gal(K/K) = (1). If K is a regular extension of some subfield E, then S(G(E)) identifies naturally with a substructure of S(G(K)), as the restriction map G(K) → G(E) is onto (if L is a finite Galois extension of E, there is a natural isomorphism between Gal(L/E) and Gal(KL/K)).

(1.5) Theorem (Interpretability of S(G(K)) in K). Let δ(ξ_{1}, …, ξ_{n}) be an L_{G}-formula, the ξ_{i}’s being of sort m, and assume that ⊧ δ(ξ_{1}, …, ξ_{n}) → ⋀_{i,j} C(ξ_{i}, ξ_{j}). There is a formula δ^{∗}(x, y) of the language of fields, where x = (x_{1}, …, x_{m}) and y = (y_{i,j})_{1≤i≤n,0≤j<m}, satisfying the following:

Let K be any field, a = (a_{1}, …, a_{m}) and b = (b_{i,j})_{1≤i≤n,0≤j<m} tuples in K. Assume that L is a Galois extension of K of degree m, generated over K by a root α of the irreducible polynomial X^{m} + a_{1}X^{m−1} + ⋯ + a_{m} = 0, and that σ_{1}, …, σ_{n} ∈ Gal(L/K) are defined by σ_{i}(α) = ∑_{j=0}^{m−1} b_{i,j}α^{j}. Then

S(G(K)) ⊧ δ(σ_{1}, …, σ_{n}) ⟺ K ⊧ δ^{∗}(a, b).

(1.6) Theorem (Elementary invariants of PAC fields). Let K_{1} and K_{2} be PAC fields, separable over a common subfield E. The following conditions are equivalent:

(1) K_{1} ≡_{E} K_{2} (K_{1} and K_{2} satisfy the same sentences in the language L(E) obtained by adding to L constant symbols for the elements of E).

(2) (i) K_{1} and K_{2} have the same degree of imperfection.

(ii) There is φ ∈ G(E) such that φ(K_{1} ∩ E^{s}) = K_{2} ∩ E^{s}, and such that (iii) if Φ : G(K_{2} ∩ E^{s}) → G(K_{1} ∩ E^{s}) is the isomorphism of profinite groups induced by φ, and SΦ : S(G(K_{1} ∩E^{s})) → S(G(K_{2} ∩E^{s})) is the dual map, then the partial map SΦ : S(G(K_{1})) → S(G(K_{2})) (with domain S(G(K_{1} ∩ E^{s}))) is an elementary L_{G}-map.

(1.7) Description of elementary (partial) isomorphisms. Theorem (1.6) yields immediately the following: Corollary. Let E_{1} ⊆ K_{1}, E_{2} ⊆ K_{2} be fields, with K_{i} a PAC field which is a separable extension of E_{i} for i = 1, 2. Assume that ψ : E_{1} → E_{2} is an L-isomorphism. The following conditions are equivalent:

(1) The partial L-map ψ : K_{1} → K_{2} is elementary.

(2) ψ extends to an isomorphism φ : E_{1}^{alg} → E_{2}^{alg} such that:

(i) K_{1} and K_{2} have the same degree of imperfection.

(ii) φ(K_{1} ∩ E_{1}^{s}) = K_{2} ∩ E_{2}^{s}.

(iii) Let Φ : G(K_{2} ∩E_{2}^{s}) → G(K_{1} ∩E_{1}^{s}) be the isomorphism of profinite groups induced by φ, and SΦ : S(G(K_{1} ∩ E_{1}^{s})) → S(G(K_{2} ∩ E_{2}^{s})) its dual. Then the partial map SΦ : S(G(K_{1})) → S(G(K_{2})) (with domain S(G(K_{1} ∩ E_{1}^{s}))) is an elementary L_{G}-map.

Remark. In case K_{1} and K_{2} are perfect, condition (i) is automatically satisfied. One can then remove the hypothesis that K_{1} and K_{2} be separable over E_{1}, E_{2}. Indeed, ψ extends uniquely to the perfect hull E_{1}^{1/p^{∞}} of E_{1}, which is contained in K_{1}; moreover, every finite Galois extension of E_{1}^{1/p^{∞}} is the composite of E_{1}^{1/p^{∞}} and of a finite Galois extension of E_{1}, so that G(K_{1} ∩ E_{1}^{s}) and G(K_{1} ∩ E_{1}^{alg}) are naturally isomorphic.

(1.8) Encoding of finitely generated extensions in rings. Let R be a ring, p(X) = X^{n}+a_{1}X^{n−1}+⋯+a_{n} a polynomial over R, and consider the R-algebra S = R[X]/(p(X)). Let us denote the image of X in S by c, so that we can identify S with the free R-module on {1, c, …, c^{n−1}}. Multiplication (in S) by the element c is then the linear trasnformation with matrix

M_{c} = ( \begin{matrix}0 & 0 & ⋯ & 0 & −a_{n} \\ 1 & 0 & ⋯ & 0 & −a_{n−1} \\ ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & ⋯ & 1 & −a_{1} \end{matrix} )

and multiplication by c^{i} is given by the matrix M_{c}^{i}. This means that we can interpret, uniformly in the parameters (a_{1}, …, a_{n}), the algebra S inside R. Moreover, assume that we are given an element c′ ∈ S satisfying p(c′) = 0. Then the endomorphism of the R-module S which sends c^{j} to c′^{j} for j = 0, …, n, is a ring endomorphism, and is interpretable in R, uniformly in the coordinates of the element c′, as is its action on S.

It is clear from the interpretation, that given elements b_{1}, …, b_{m} of S, with coordinates (b_{i,j}) with respect to the basis {1, c, …, c^{n−1}}, any statement (in S) about b_{1}, …, b_{m} which is expressible by polynomial equations, is equivalent to a system of polynomial equations in the tuple (ā, (b_{i,j})) (in R).

2. Proof of the result

(2.1) Definition. We call an L-formula φ(x̄; ȳ) basic (over x̄), if it can be written α(x̄, ȳ)∧ β(x̄, ȳ) ∧ γ(x̄, ȳ) for some L-formulas α, β and γ, and if there are polynomials over ℤ

— F̄(X̄, Ȳ) (F̄ = (F_{1}, …, F_{ℓ})), g(X̄, Ȳ),

— p(X̄, Ȳ , Z) monic and of degree n in Z, q(X̄, T) monic and of degree m in T, Q(X̄, Ȳ , Z, T) monic and of degree n/m in Z,

— s(X̄, Ȳ , Z), r_{2}(X̄, Ȳ , Z), …, r_{n}(X̄, Ȳ , Z) of degree < n in Z, s_{2}(X̄, T), …, s_{m}(X̄, T) of degree < m in T, r_{1}(X̄, Ȳ , Z) = Z, s_{1}(X̄, T) = T,

— a map μ : {1, …, n}×{1, …, n} → {1, …, n}, and a surjective map π : {1, …, n} → {1, …, m}, satisfying the following conditions:

(A) The formula α(x̄, ȳ) says that the polynomial p(x̄, ȳ, Z) in Z is irreducible over K, and the polynomial q(x̄, T) in T is irreducible over K.

(B) The formula β(x̄, ȳ) says that (x̄, ȳ) belongs to the algebraic set V defined by the equations F̄(X̄, Ȳ) = 0, that g(x̄, ȳ) ≠ 0. We assume that F̄ and g are such that, if (x̄, ȳ) satisfies β(x̄, ȳ), then the algebraic set V (x̄) defined by F̄(x̄, Ȳ) = 0 is absolutely irreducible and that if t satisfies q(x̄, T) = 0, then the algebraic set U(x̄, t) defined by F̄(x̄, Ȳ) = 0 and Q(x̄, Ȳ , Z, t) = 0 is absolutely irreducible and projects generically onto V (x̄).

Let R be the subring of K generated by x̄, ȳ, let S = R[Z]/(p(x̄, ȳ, Z)), and denote by c the image of Z in S. We require furthermore that:

(1) The elements r_{2}(x̄, ȳ, c), …, r_{n}(x̄, ȳ, c) satisfy p(x̄, ȳ, Z) = 0. For 1 ≤ i < j ≤ n, the elements (r_{i}(x̄, ȳ, c) − r_{j}(x̄, ȳ, c)) are invertible in S_{1} = S[g(x̄, ȳ)^{−1}].

(2) The elements d = s(x̄, ȳ, c), d_{i} = s_{i}(x̄, d), i = 1, …, m, satisfy q(x̄, T) = 0.

(3) Q(x̄, ȳ, c, d) = 0.

(4) Define σ_{i}(x̄, ȳ) to be the R-linear endomorphism of S obtained by sending c^{j} to r_{i}(x̄, ȳ, c)^{j} for i = 1, …, n, j = 0, …, n − 1. For i, j ∈ {1, …, n}, σ_{i}(x̄, ȳ) ∘ σ_{j}(x̄, ȳ) = σ_{μ(i,j)}(x̄, ȳ).

(5) For i = 1, …, n, σ_{i}(x̄, ȳ)(d) = s_{π(i)}(x̄, d) (or equivalently, s(x̄, ȳ, r_{i}(x̄, ȳ, d)) = s_{π(i)}(x̄, d)).

(C) γ(x̄, ȳ) is of the form δ^{∗}, for some δ(σ_{1}(x̄, ȳ), …, σ_{n}(x̄, ȳ)), with δ ∈ L_{G}, and the σ_{i}(x̄, ȳ) the elements defined by the formulas given in (B)(4). Note that (B)(4) already describes the isomorphism type of the group {σ_{1}(x̄, ȳ), …, σ_{n}(x̄, ȳ)}.

(2.2) Definition. A formula φ(x̄; ȳ) is semi-basic (over x̄), if there is a basic formula φ_{1}(x̄, x̄_{1}; ȳ, ȳ_{1}) such that

— φ(x̄; ȳ) = ∃x̄_{1}ȳ_{1} φ_{1}(x̄, x̄_{1}; ȳ, ȳ_{1})

— Write φ_{1} = α_{1} ∧ β_{1} ∧ γ_{1} as in the definition of basic. If (ā, ā_{1}, b̄, b̄_{1}) are any tuples in a field K which satisfy β_{1}, then ā_{1} ∈ k(ā)^{alg}, and b̄_{1} ∈ k(ā, b̄)^{alg}, where k is the prime subfield of K.

(2.3) Remarks and comments. (1) Our definition is independent of the characteristic of the field considered and does not impose that the algebraic set V introduced in (B) be irreducible. In particular we allow constant polynomials to appear among the defining polynomials of V. Recall that the notion of absolute irreducibility of an algebraic set is elementary in the parameters defining this set, and in fact is expressible by quantifier-free formulas (see e.g. [DS] (1.11) and (2.10)). If (x̄, ȳ) is a generic of V and V (x̄) is irreducible, then there is a Zariski open subset U of the projection of V on X̄-space, such that V (x̄) will be absolutely irreducible for all x̄ ∈ U. Using the remark made in (1.8), it is also clear that items (1) – (5) of (B) can be expressed by systems of polynomial equations in x̄, ȳ, so that they will be consequence of the fact that certain polynomials belong to the ideal generated by F̄(X̄, Ȳ).

(2) If one fixes a complete theory of perfect Frobenius fields, then one only needs to consider semi-basic formulas with no subformula of type γ: this is because an isomorphism between two finite quotients of the absolute Galois group of a model yields an elementary L_{G}-map of the associated inverse system, see [CDM], or [FJ] 23.21 for a proof.

(3) The separation of the variables of a semi-basic formula in two kinds is for specialisation purposes: we intend to describe uniformly sets defined in a perfect PAC field K by the formula φ(ā; ȳ), where ā is a tuple of K.

(2.4) Lemma. Let T be the theory of perfect PAC fields, let K be a perfect PAC field, (ā, b̄) be a tuple in K, and consider the set Σ(x̄, ȳ) of all semi-basic formulas (over x̄) satisfied by (ā, b̄) in K. Then T ∪Σ(x̄, ȳ) is complete, i.e., if K_{2} is a model of T, and (ā′, b̄′) is a tuple of elements of K_{2} which satisfies Σ(x̄, ȳ), then the partial map φ : K → K_{2} which sends (ā, b̄) to (ā′, b̄′) is L-elementary. Proof. Let K_{2} be a perfect PAC field and (ā′, b̄′) be a tuple of K_{2} satisfying Σ(x̄, ȳ). Then K and K_{2} have the same prime subfield k. We want to show that there is a k-isomorphism φ : k(ā, b̄)^{alg} → k(ā′, b̄′)^{alg} satisfying the requirements of Corollary (1.7) (see also Remark (1.7)), and sending ā to ā′, b̄ to b̄′.

First of all, note that our assumption implies that the map φ_{0} sending (ā, b̄) to (ā′, b̄′) defines an isomorphism φ_{0} : k(ā, b̄) → k(ā′, b̄′), i.e., that Σ(x̄, ȳ) describes the isomorphism type of k(ā, b̄): indeed, let ā_{1} ∈ k(ā)^{alg} ∩K be such that k(ā, ā_{1}, b̄) is a regular extension of k(ā, ā_{1}), and let V be the locus of (ā, ā_{1}, b̄) over k. Then V (ā, ā_{1}) is absolutely irreducible, and our assumption implies that there is ā′_{1} in k(ā′)^{alg} ∩ K_{2} such that (ā′, ā′_{1}, b̄′) satisfies the same quantifier-free formulas as (ā, ā_{1}, b̄). Hence (ā′, ā′_{1}, b̄′) is a generic of V, and φ_{0} is an isomorphism.

By compactness, it suffices to show that for every finite algebraic extension L of k(ā, b̄), such that L is Galois over L ∩ K of degree n, there is an extension φ_{L} of φ_{0} to L which satisfies the restrictions to L of the requirements of (1.6) and (1.7), i.e. ((i) is vacuous because K and K_{2} are perfect):

(ii) φ_{L}(L ∩ K) = φ_{L}(L) ∩ K_{2}. (iii) The map double dual of φ_{L}, SΦ_{L} : Gal(L/L ∩ K) → Gal(φ_{L}(L)/φ_{L}(L) ∩ K_{2}), is an

elementary partial L_{G}-map S(G(K)) → S(G(K_{2})).

We set B = k(ā, b̄)^{alg} ∩ K, A = k(ā)^{alg} ∩ K. Let L be a finite algebraic extension of k(ā, b̄), with L Galois over L ∩ K and of degree n. Then L and k(ā)^{alg} are linearly disjoint over their intersection L_{0}, so that L is a regular extension of L_{0}. Choose b̄_{1} ∈ B such that L ∩ K = k(ā, b̄, b̄_{1}), then choose ā_{1} ∈ A such that k(ā, ā_{1}, b̄, b̄_{1}) is a regular extension of k(ā, ā_{1}), and k(ā, ā_{1}) contains A ∩ L = A ∩ L_{0}. Replacing L by L(ā_{1}), we may assume that L contains ā_{1}, so that B ∩ L = k(ā, ā_{1}, b̄, b̄_{1}) and L_{0} = A ∩ L = k(ā, ā_{1}).

Let R_{0} be the prime subring of k (so R_{0} = 𝔽_{p} = k, or R_{0} = ℤ). Choose a generator d of L_{0} over k(ā, ā_{1}) and a generator c of L over k(ā, ā_{1}, b̄, b̄_{1}). Enlarging ā_{1} and b̄_{1} if necessary, we may assume that

— the minimal monic polynomial of d over k(ā, ā_{1}) has its coefficients in R_{0}[ā, ā_{1}], the conjugates of d over k(ā, ā_{1}) are in R_{0}[ā, ā_{1}, d], and d ∈ R_{0}[ā, ā_{1}, b̄, b̄_{1}, c].

— the minimal monic polynomial of c over k(ā, ā_{1}, b̄, b̄_{1}) has its coefficients in R_{0}[ā, ā_{1}, b̄, b̄_{1}], the minimal monic polynomial of c over k(ā, ā_{1}, b̄, b̄_{1}, d) has its coefficients in R_{0}[ā, ā_{1}, b̄, b̄_{1}, d], the conjugates of c over k(ā, ā_{1}, b̄, b̄_{1}) are all in R_{0}[ā, ā_{1}, b̄, b̄_{1}, c], and the difference of any two distinct conjugates is invertible in R_{0}[ā, ā_{1}, b̄, b̄_{1}, c].

Hence, there exist integral polynomials

— p(X̄, X̄_{1}, Ȳ , Ȳ_{1}, Z) monic and of degree n in Z, q(X̄_{1}, Ȳ_{1}, T) monic and of degree m = [L_{0} : K] in T, Q(X̄, X̄_{1}, Ȳ , Ȳ_{1}, Z, T) monic and of degree n/m in Z,

— s(X̄, X̄_{1}, Ȳ , Ȳ_{1}, Z), r_{2}(X̄, X̄_{1}, Ȳ , Ȳ_{1}, Z), …, r_{n}(X̄, X̄_{1}, Ȳ , Ȳ_{1}, Z) of degree < n in Z, s_{2}(X̄, X̄_{1}, T), …, s_{m}(X̄, X̄_{1}, T) of degree < m in T, r_{1}(X̄, X̄_{1}, Ȳ , Ȳ_{1}, Z) = Z, s_{1}(X̄, X̄_{1}, T) = T, u_{i,j}(X̄, X̄_{1}, Ȳ , Ȳ_{1}, Z) of degree < n in Z for 1 ≤ i < j ≤ n,

such that:

— p(ā, ā_{1}, b̄, b̄_{1}, Z) is the minimal polynomial of c over K, q(ā, ā_{1}, T) is the minimal polynomial of d over K, Q(ā, ā_{1}, b̄, b̄_{1}, Z, d) is the minimal polynomial of c over K(d).

— the conjugates c_{1} = c, c_{2}, …, c_{n} of c over K are given by c_{i} = r_{i}(ā, ā_{1}, b̄, b̄_{1}, c) for i = 1, …, n, and u_{i,j}(ā, ā_{1}, b̄, b̄_{1}, c)(c_{i} − c_{j}) = 1 for 1 ≤ i < j ≤ n.

— d = s(ā, ā_{1}, b̄, b̄_{1}, c).

— the conjugates d_{1} = d, d_{2}, …, d_{m} of d over K are given by d_{i} = s_{i}(ā, ā_{1}, d).

Let σ_{i} ∈ Gal(L/B ∩ L) be the element sending c to c_{i} for i = 1, …, n. Let μ : {1, …, n} × {1, …, n} → {1, …, n} be defined by σ_{i} ∘ σ_{j} = σ_{μ(i,j)}. Because the σ_{i}’s permute the d_{j}’s, there is a surjection π : {1, …, n} → {1, …, m} such that for every i we have σ_{i}(d) = d_{π(i)}.

By construction, k(ā, ā_{1}, b̄, b̄_{1}) is a regular extension of k(ā, ā_{1}), and L = k(ā, ā_{1}, b̄, b̄_{1}, c) is a regular extension of L_{0} = k(ā, ā_{1}, d). Hence, letting V be the algebraic locus of (ā, ā_{1}, b̄, b̄_{1}), and U the algebraic locus of (ā, ā_{1}, b̄, b̄_{1}, c, d), we obtain that the algebraic sets V (ā, ā_{1}) = {(Ȳ , Ȳ_{1}) | (ā, ā_{1}, Ȳ , Ȳ_{1}) ∈ V } and U(ā, ā_{1}, d) = {(Ȳ , Ȳ_{1}, Z) | (ā, ā_{1}, Ȳ , Ȳ_{1}) ∈ V, Q(ā, ā_{1}, Ȳ , Ȳ_{1}, Z, d) = 0} are absolutely irreducible, with U(ā, ā_{1}, d) projecting generically onto V (ā, ā_{1}). Our requirements on the c_{i}’s and d_{j}’s give us that there is a basic formula (satisfied by (ā, ā_{1}, b̄, b̄_{1})) containing (x̄, x̄_{1}, ȳ, ȳ_{1}) ∈ V and which describes the Galois extension L as well as the action of Gal(L/K) over L, see (1.8). Moreover, any L_{G}- formula satisfied by (σ_{1}, …, σ_{n}) is implied by some basic formula satisfied by (ā, ā_{1}, b̄, b̄_{1}), see (1.5).

Using the fact that ā_{1}, b̄_{1} are algebraic over k(ā, b̄), and that (ā′, b̄′) satisfy Σ(x̄, ȳ), there are tuples ā′_{1} ∈ k(ā′)^{alg} ∩ K_{2}, and b̄′_{1} ∈ k(b̄′)^{alg} ∩ K_{2}, such that (ā, ā_{1}, b̄, b̄_{1}) and (ā′, ā′_{1}, b̄′, b̄′_{1}) satisfy the same basic formulas. Then the polynomials p(ā′, ā′_{1}, b̄, b̄′_{1}, Z), q(ā′, ā′_{1}, T) are irreducible over K_{2}.

Let c′ ∈ K_{2}^{s} be a root of p(ā′, ā′_{1}, b̄′, b̄′_{1}, Z), and let c′_{i} = r_{i}(ā′, ā′_{1}, b̄′, b̄′_{1}, c′) for i = 2, …, n, d′ = s(ā′, ā′_{1}, b̄′, b̄′_{1}, c′). We then define φ_{L} by sending (ā_{1}, b̄_{1}, c) to (ā′_{1}, b̄′_{1}, c′). Then φ_{L}(c_{i}) = c′_{i}, φ_{L}(d) = d′, and SΦ_{L}(σ_{i}) is the automorphism of L′ = φ_{L}(L) over k(ā′, ā′_{1}, b̄′, b̄′_{1}) which sends c′ to c′_{i}. Then φ_{L} satisfies the requirements (ii) and (iii).

(2.5) Corollary. Let φ(x̄, ȳ) be a formula. Then φ(x̄, ȳ) is equivalent, modulo T, to a finite disjunction of semi-basic formulas over x̄.

Proof. This is an easy application of the compactness theorem (see e.g. Lemma 3.2.1 in [CK], applied to the set Δ of finite disjunctions of formulas ψ(x̄, ȳ) which are either in T or semi-basic.)

(2.6) Reduction of the problem. Let K be a perfect PAC field. Lemma (5.3) of [ChDM] shows that if φ(x̄, ȳ) is a formula, ā is a tuple of elements of K and the set defined by φ(ā, ȳ) in K is finite, then this set is contained in the field theoretic algebraic closure of the subfield of K generated by ā (in other words: a subfield A of K is model-theoretically algebraically closed if and only if it is relatively field-theoretically algebraically closed in K). By Lemma (5.6) of [ChDM], it is therefore enough to show that for any formula φ(x̄, ȳ), there is an integer N such that, for any model K of T and tuple ā in K, the set defined by φ(ā, ȳ) in K is finite if and only if it is of cardinality ≤ N.

Let φ(x̄, ȳ) be a formula. By Corollary (2.5), there are semi-basic formulas φ_{i}(x̄; ȳ), i = 1, …, r, such that T ⊢ φ(x̄, ȳ) ↔ ⋁_{i} φ_{i}(x̄; ȳ). Clearly, in any perfect PAC field K, the set defined by φ(ā, ȳ) will be finite if and only if each of the sets defined by the formulas φ_{i}(ā; ȳ) are finite. It therefore suffices to find a uniform bound N for each of the formulas φ_{i}(x̄; ȳ).

Let φ(x̄; ȳ) be a semi-basic formula, and write it ∃x̄_{1}, ȳ_{1} φ_{1}(x̄, x̄_{1}; ȳ, ȳ_{1}), with φ_{1} satisfying the requirements of the definition (2.2). Then, in the notation of (2.2), our assumption on β_{1} and the fact that the theory of algebraically closed fields is algebraically bounded ([D]) imply that there is an integer M such that, for any field K and tuple (ā, b̄) in K, the set of tuples satisfying β_{1}(ā, x̄_{1}, b̄, ȳ_{1}) has size ≤ M. This reduces the problem to finding a uniform bound for basic formulas. The next lemma shows that in that case, the bound is 1.

(2.7) Proposition. Let φ(x̄; ȳ) be a basic formula (over x̄). We use the notation of (2.1). Let K be a perfect PAC field, ā a tuple of elements of K, and assume that the set D(ā) defined by φ(ā; ȳ) is non-empty. Then D(ā) is finite if and only if |V (ā)| = 1, if and only if dim(V (ā)) = 0.

Proof. The second equivalence is clear, since V (ā) is assumed to be absolutely irreducible. The right-to-left direction of the first equivalence is also clear. Let us now assume that V (ā) is infinite, we will show that there are infinitely many tuples b̄ satisfying φ(ā; ȳ).

Fix b̄_{0} ∈ D(ā), and let c_{0} be a root of p(ā, b̄_{0}, Z), and L_{0} = K(c_{0}), d = s(ā, b̄_{0}, c_{0}). Let σ_{i} ∈ Gal(L_{0}/K) be defined by σ_{i}(c_{0}) = r_{i}(ā, b̄_{0}, c_{0}) for i = 1, …, n. Since (ā, b̄_{0}) satisfy φ, the polynomial q(ā, T) is irreducible over K, and the set U(ā, d) is absolutely irreducible.

Let b̄ be a generic of V (ā) over K. Then (ā, b̄) satisfies β(x̄, ȳ). As b̄ is a generic of V (ā), the absolute irreducibility of U(ā, d) implies that the monic polynomial Q(ā, b̄, Z, d) is irreducible over K(b̄, d) (and of degree n/m). Hence, if c is a root of Q(ā, b̄, Z, d), we have [K(b̄, c) : K(b̄)] = n, c is a root of the degree n polynomial p(ā, b̄, Z), so that p(ā, b̄, Z) is the minimal polynomial of c over K(b̄), and in particular is irreducible.

We also have d = s(ā, b̄, c), K(b̄, c) is a regular extension of K(d), the elements c_{i} = r_{i}(ā, b̄, c), i = 1, …, n, are the conjugates of c over K(b̄), and if σ′_{i} ∈ Gal(K(b̄, c)/K(b̄)) is defined by σ′_{i}(c) = c_{i} for i = 1, …, n, then σ′_{i}(d) = s_{π(i)}(ā, d) = σ_{i}(d), and σ′_{i}σ′_{j} = σ′_{μ(i,j)} for i, j ∈ {1, …, n}.

Because K(b̄, c) is a regular extension of K(d), we may identify Gal(K(b̄, c, c_{0})/K(b̄)) with the subgroup of Gal(K(b̄, c)/K(b̄)) × Gal(K(c_{0})/K) consisting of pairs (σ′, σ) whose restrictions to K(d) agree. By the remarks made in the previous paragraph, the set H = {(σ′_{1}, σ_{1}), …, (σ′_{n}, σ_{n})} is a subgroup of Gal(K(b̄, c, c_{0})/K(b̄)), which projects onto Gal(K(b̄, c)/K(b̄)) and onto Gal(K(c_{0})/K). Hence the subfield M of K(b̄, c, c_{0}) fixed by H is a regular extension of K of degree n, M(c) = M(c_{0}), and Gal(MK^{alg}/M) is isomorphic to G(K).

Thus, (see e.g. 12.11 of [FJ]), there is a perfect PAC field F containing M, and which is a regular extension of M. Then G(F) projects on G(M), which projects onto G(K) (via the restriction maps). Since K is PAC, we know that G(K) is projective ([FJ], 10.17), and that there is a closed subgroup G_{1} of G(F) which lifts G(K), i.e., such that the restriction to K^{alg} defines an isomorphism between G_{1} and G(K) (see 20.11 in [FJ]). Let K_{1} be the subfield of F^{alg} fixed by G_{1}. Then K_{1} is an algebraic extension of F, and therefore is PAC ([FJ], 10.7). Moreover, G(K_{1}) = G_{1}, so that K_{1}K^{alg} = K_{1}^{alg}, and K_{1} is a regular extension of K containing M. Theorem (1.6) gives us then that K ≺ K_{1}.

It remains to show that (ā, b̄) satisfies α∧β ∧γ in K_{1}. From M(c) = M(c_{0}), we obtain K_{1}(c) = K_{1}(c_{0}), and therefore (ā, b̄) satisfies α. It certainly satisfies β. Our choice of H and of M implies that the elements σ_{i} and σ′_{i} define the same element of Gal(K_{1}(c_{0})/K_{1}). Hence they satisfy the same L_{G}-formulas in SG(K_{1}), and therefore (ā, b̄) satisfies γ because (ā, b̄_{0}) does. This implies that (ā, b̄) satisfies φ. Hence K has an elementary extension K_{1} in which the set D(ā) defined by φ(ā; ȳ) has a new point (namely b̄). This implies that D(ā) is infinite, and finishes the proof.

Remark. The proof shows that V (ā) is the Zariski closure of D(ā). Thus we have shown the following: let φ(x̄; ȳ) be a semi-basic formula over x̄, and write it as in Definition (2.2)

φ(x̄; ȳ) = ∃x̄_{1}, ȳ_{1} (α_{1} ∧ β_{1} ∧ γ_{1})(x̄, x̄_{1}, ȳ, ȳ_{1}).

Let K be a PAC field, ā a tuple of elements of K. Then either the set defined by φ(ā; ȳ) in K is empty, or for some tuple ā_{1} in K (which is field-algebraic over ā), its Zariski closure coincides with the Zariski closure of the projection of the constructible set defined by β_{1}(ā, ā_{1}, ȳ, ȳ_{1}) on the ȳ-plane. In particular both sets have the same dimension.

(2.8) Theorem. The theory T of perfect PAC fields is algebraically bounded.

Proof. Clear from the discussion in (2.6) and Proposition (2.7).

Bibliography.

[CK] C.C. Chang, H.J. Keisler, Model theory, North-Holland, Amsterdam 1977.

[C] Z. Chatzidakis, Properties of forking in ω-free pseudo-algebraically closed fields, to

appear in J. of Symb. Logic.

[ChDM] Z. Chatzidakis, L. van den Dries, A. Macintyre, Definable sets over finite fields, J. reine u. ang. Math. 427 (1992), 107 – 135.

[CDM] G. Cherlin, L. van den Dries, A. Macintyre, Decidability and Undecidability Theorems for PAC-Fields, Bull.AMS 4 (1981), 101-104.

[CDM1] G. Cherlin, L. van den Dries, A. Macintyre, The elementary theory of regularly closed fields, preprint (1980).

[CDM2] G. Cherlin, L. van den Dries, A. Macintyre, The elementary theory of regularly closed

fields, preprint (1982).

[D] L. van den Dries, Dimension of definable sets, algebraic boundedness and henselian fields, Annals of Pure and Applied Logic 45 (1989), 189 – 209.

[DS] L. van den Dries, K. Schmidt, Bounds in the theory of polynomials rings over fields. A non-standard approach, Invent. Math. 76 (1984), 77 – 91.

[FJ] M. Fried, M. Jarden, Field Arithmetic, Ergebnisse 11, Springer Berlin-Heidelberg 1986.

[J] M. Jarden, Algebraic dimension over Frobenius fields Forum Math. 6 (1994), 43-63.

[L] S. Lang, Introduction to algebraic geometry, Addison-Wesley Pub. Co., Menlo Park 1973.

[M] D. Marker, Model Theory: An Introduction, GTM 217, Springer-Verlag, 2002.

[RZ] L. Ribes, P. Zalesskii, Profinite groups, Ergebnisse 40, Springer-Verlag Berlin Heidelberg 2000.

Current addresses:

UFR de Mathématiques Université Paris 7, Case 7012 2, place Jussieu 75251 Paris Cedex 05 France e-mail: zoe@logique.jussieu.fr

Institute of Mathematics The Hebrew University Givat Ram 91904 Jerusalem Israel e-mail: ehud@math.huji.ac.il