ON THE COMPLEXITY OF THE ISOMORPHISM RELATION FOR FIELDS OF FINITE TRANSCENDENCE DEGREE

SIMON THOMAS AND BOBAN VELICKOVIC

Abstract. Confirming a conjecture of Hjorth and Kechris [HK], we prove that the isomorphism relation for fields of finite transcendence degree is a universal essentially countable Borel equivalence relation. We also prove that the theory of fields of finite transcendence degree does not admit canonical models.

1. Introduction

Given a class K of structures for a fixed first-order language L, it is natural to try to understand the complexity of the isomorphism relation for K. For those classes consisting of the countable models of some L_{ω_{1},ω}-sentence, Friedman and Stanley [FS] proposed to use the methods of descriptive set theory to study their isomorphism relations. Hjorth and Kechris [HK] continued this project and situated it within the general theory of Borel equivalence relations. This provided tools for the analysis of the isomorphism relation and generated a host of interesting open problems. The main result of this paper is that the isomorphism relation for fields of finite transcendence degree is as complex as it conceivably could be. (The analogous result for arbitrary countable fields was proved independently in [FK] and [FS], where it was shown that the associated isomorphism relation is complete analytic.)

Before we can give an exact statement of our main result, we first need to describe how to represent the class of countably infinite structures of a given first order language by the elements of a Polish space. For convenience, we shall work only with relational first order languages. If there are function symbols, as is the case with the language of field theory, then we replace them by relation symbols representing the graphs of the associated functions. Given a language L = {R_{i} | i ∈ I}, where

The research of the first author was partially supported by NSF Grants.

2 SIMON THOMAS AND BOBAN VELICKOVIC

R_{i} is an n_{i}-ary relation symbol, let

X_{L} = ∏_{i∈I} 2^{ℕ^{n_{i}}} .

Then the elements of X_{L} code L-structures with universe ℕ as follows. Given x = (x_{i})_{i∈I} ∈ X_{L}, the structure

M_{x} = ⟨ℕ, R_{i}^{x}⟩_{i∈I}

represented by x is defined by:

R_{i}^{x}(k_{1}, …, k_{n_{i}}) ⇔ x_{i}(k_{1}, …, x_{n_{i}}) = 1.

The symmetric group S_{∞} on ℕ is a Polish group with the usual product topology inherited from ℕ^{ℕ}; and X_{L} together with the natural action of S_{∞} is a Polish S_{∞}- space. Notice that x, y ∈ X_{L} lie in the same S_{∞}-orbit iff M_{x} ≅ M_{y}. For this reason, the action is usually called the logic action of S_{∞} on X_{L}. Given an L_{ω_{1},ω} sentence σ,

Mod(σ) = {x ∈ X_{L} | M_{x} ⊧ σ}

represents the class of all countably infinite models of σ. Note that Mod(σ) is an S_{∞}-invariant Borel subset of X_{L}. We shall denote the restriction of the isomorphism relation ≅ to Mod(σ) by ≅_{σ}. When working with Mod(σ), we shall usually identify M_{x} with x.

The notion of Borel reducibility will enable us to discuss the complexity of the isomorphism relation ≅_{σ}. Suppose that X and Y are Borel sets and that E and F are equivalence relations on X and Y respectively. Then E is Borel reducible to F, written E ≤_{B} F if there exists a Borel function f : X → Y such that xEy ⇔ f(x)F f(y). We say that E and F are Borel bireducible and write E ∼_{B} F if E ≤_{B} F and F ≤_{B} E. We say that E is smooth if there exists a Polish space Y such that E ≤_{B} Δ(Y), where Δ(Y) is the equality relation on Y. If ≅_{σ} is smooth, then we say that the models of σ are concretely classifiable.

In general, if σ is an arbitrary sentence of L_{ω_{1},ω}, then the isomorphism relation ≅_{σ} is analytic rather than Borel. (For example, the isomorphism relation for arbitrary countable fields is complete analytic. See either [FK] or [FS].) But there is a large class of theories σ for which ≅_{σ} is a Borel relation. In particular, ≅_{σ} is Borel for those theories σ whose models have “finite rank” in a broad sense, such as: the

FIELDS OF FINITE TRANSCENDENCE DEGREE 3

theory of finitely generated groups; the theory of finite rank torsion-free abelian groups; the theory of fields of finite transcendence degree, etc. In order to study these examples, it is useful to introduce the notion of an essentially countable equivalence relation. An equivalence relation E is said to be countable if each equivalence class of E is countable. Given a theory σ, we say that ≅_{σ} is essentially countable iff ≅_{σ}≤_{B} E for some countable Borel equivalence relation E on a Borel set X. In this case, Kechris [Ke1] has shown that there exists a countable Borel equivalence relation F such that ≅_{σ}∼_{B} F. There is a natural model-theoretic characterisation of essentially countable theories; namely, Hjorth and Kechris [HK] have shown that the following are equivalent:

(i) ≅_{σ} is essentially countable; (ii) there is a countable fragment F ⊆ L_{ω_{1},ω} containing σ such that if M = ⟨ℕ, −⟩

is a model of σ, there exists ā ∈ ℕ^{<ω} such that Th_{F} (⟨M, ā⟩) is ℵ_{0}-categorical. ( Condition (ii) can be interpreted as saying that the models of σ have “finite rank” in some sense.) For such theories σ, we can use the theory of countable Borel equivalence relations to study the complexity of the isomorphism relation ≅_{σ}. (For the general theory of such equivalence relations, see [JKL].)

The least complex countable Borel equivalence relations are those which are smooth. It turns out that there is also a most complex countable Borel equivalence relation. Define a countable Borel equivalence relation E to be universal if F ≤_{B} E for any other countable Borel equivalence relation F. Such an equivalence relation exists, and it is obviously unique up to Borel bireducibility. One particular realization of it is the orbit equivalence relation given by the shift action of the free group F_{2} on two generators on 2^{F_{2}}. Recall that given any set X and any group G, the shift action of G on X^{G} is given by

x(h)g = x(hg).

Let E(F_{2}, 2^{F_{2}}) denote the orbit equivalence relation induced by the shift action of the free group on two generators F_{2} on 2^{F_{2}}. In [DJK], it was shown that E(F_{2}, 2^{F_{2}}) is a universal countable Borel equivalence relation. If ≅_{σ}∼_{B} E_{E}(F_{2}, 2^{F_{2}}), then we say that ≅_{σ} is a universal essentially countable Borel equivalence relation. For example, in [JKL], it was shown that the isomorphism relation for locally finite trees is a universal essentially countable Borel equivalence relation; and in [TV], we

4 SIMON THOMAS AND BOBAN VELICKOVIC

proved that the isomorphism relation for finitely generated groups is also a universal essentially countable Borel equivalence relation. The main result of this paper is that the isomorphism relation for fields of finite transcendence degree is a universal essentially countable Borel equivalence relation. This confirms Conjecture 5.9 of [HK].

We shall also prove that the theory of fields of finite transcendence degree does not admit canonical models. Here a sentence σ of L_{ω_{1},ω} is said to admit canonical models if there exists an “effectively definable” function f which to each countable model M ∈ Mod(σ) assigns a countable model f(M) of σ such that

M ≅ N implies f(M) = f(N) ≅ M.

We usually cannot expect to be able to choose f so that the structure f(M) also has universe ℕ; since Burgess has shown that for any sentence σ of L_{ω_{1},ω}, the following statements are equivalent:

(i) Mod(σ) is concretely classifiable; (ii) there exists a Borel function f : Mod(σ) → Mod(σ) such that M ≅ N implies

f(M) = f(N) ≅ M. However, in [HK], Hjorth and Kechris showed that there are many interesting theories which are not concretely classifiable and yet do admit canonical models, including the theories of rank 1 abelian torsion-free groups and of rigid locally finite trees; and they also proved that the theory of locally finite trees does not admit canonical models. In [TV], we proved that the theory of finitely generated groups does not admit canonical models.

It is quite difficult to give a precise formulation of the notion of “effectively definable” in the case when Mod(σ) is not concretely classifiable. An exact definition can be found in Section 7 of [HK]; but some set-theoretic sophistication is necessary in order to understand this definition. Fortunately Hjorth and Kechris have found the following algebraic property which is equivalent to admitting canonical models. Suppose that G is a Polish group and that a : X × G → X is a Borel action of G on the Borel space X. Let E_{a} be the associated orbit equivalence relation on X. A Borel function α : E_{a} → G is called a cocycle if whenever xE_{a}yE_{a}z, then α(x, z) = α(x, y)α(y, z). We say that the action a has the cocycle property if there

FIELDS OF FINITE TRANSCENDENCE DEGREE 5

exists a Borel cocycle α such that a(x, α(x, y)) = y. (For the basic facts concerning Polish group actions, we refer the reader to [BK] or [Ke2].) In [HK], Hjorth and Kechris proved that for any sentence σ of L_{ω_{1},ω}, the following statements are equivalent:

(i) σ admits canonical models; (ii) the logic action of S_{∞} on Mod(σ) has the cocycle property. In Section 2, we shall prove that if σ is the theory of fields of finite transcendence dimension, then the logic action of S_{∞} on Mod(σ) does not have the cocycle property. Hence the theory of fields of finite transcendence dimension does not admit canonical models.

We shall end this introduction with a few words on our group-theoretic and fieldtheoretic notation. In this paper, permutation groups always act on the right. If (G, Ω) is a permutation group, α ∈ Ω and g ∈ G, then the image of α under g will be denoted by either αg or α^{g}; and the corresponding orbit equivalence relation will be denoted by E(G, Ω). Our field-theoretic notation is standard. For example, if x_{1}, …, x_{n} is a collection of algebraically independent elements over the field F, then F [x_{1}, …, x_{n}] denotes the corresponding polynomial ring and F (x_{1}, …, x_{n}) denotes the corresponding field of rational functions. Also if F is a field, then the algebraic closure of F is denoted by \overline{F}. Finally if z ∈ ℤ ∖ \{0\}, then sgn(z) = z/|z| denotes the sign of z.

2. Fields of finite transcendence degree

In this section, we shall first prove our main result which says that the isomorphism relation for fields of finite transcendence degree is a universal essentially countable Borel equivalence relation. Then we shall prove that the theory of fields of finite transcencence degree does not admit canonical models. Our proofs are based closely on those of Fried and Kollár [FK], who showed how to code an arbitrary infinite structure M in a field of transcendence degree |M|. The following result, which will be proved in Section 3, will enable us to code the orbit equivalence relation E(F_{2}, 2^{F_{2}}) within the class of fields of finite transcendence degree.

Theorem 2.1. Let u be a transcendental element over ℚ. Then there exists a free subgroup G = ⟨a, b⟩ of the general linear group GL(3, ℚ(u)) such that if the matrices g_{1}, g_{2}, h_{1}, h_{2} ∈ G satisfy the equality

g_{1} + 3h_{1} = g_{2} + 3h_{2},

then g_{1} = g_{2} and h_{1} = h_{2}.

We shall fix such a free group G = ⟨a, b⟩ for the rest of this section. The Cayley graph of G is the labelled directed graph Γ_{G} = (G; E_{a}, E_{b}), where

E_{a} = {(g, ag) | g ∈ G} and E_{b} = {(g, bg) | g ∈ G}.

Thus G is the set of vertices of Γ_{G}; and we regard the elements of E_{a} and E_{b} as directed edges labelled by a, b respectively. For each g ∈ G, let g^{ρ} be the right translation map defined by h^{g^{ρ}} = hg for all h ∈ G. Then the following result is well-known.

Lemma 2.2. Aut Γ_{G} = {g^{ρ} | g ∈ G}.

□

Clearly the permutation group (G, 2^{G}) can be naturally identified with the permutation group (G, P(G)), where G acts on its powerset P(G) via right translations. Thus E (G, P(G)) ∼_{B} E (G, 2^{G}). For technical reasons, we shall work with P̂(G) = {C ⊆ G | |C| ≥ 2}, rather than with P(G). Of course, it is easily shown that E (G, P̂(G)) ∼_{B} E (G, P(G)).

For each C ∈ P̂(G), we shall define an associated field F_{C} of finite transcendence degree such that if C, D ∈ P̂(G), then F_{C} ≅ F_{D} iff there exists g ∈ G such that Cg = D. (Formally we are required to produce fields with underlying set ℕ. However, this will cause no difficulties, since the fields we define all have canonical isomorphisms to such fields.) To begin the construction, let F_{0} = ℚ(u), so that G ⩽ GL(3, F_{0}). Now let {x_{i}, y_{i}, z_{i} | 1 ≤ i ≤ 3} be a set of algebraically independent elements over F_{0}. Then the general linear group GL(3, F_{0}) acts in a natural way as a group of automorphisms of the polynomial ring

R = F_{0}[x_{1}, x_{2}, x_{3}, y_{1}, y_{2}, y_{3}, z_{1}, z_{2}, z_{3}].

FIELDS OF FINITE TRANSCENDENCE DEGREE 7

Namely, if φ = (α_{ij}), then

x_{i}^{φ} = α_{i1}x_{1} + α_{i2}x_{2} + α_{i3}x_{3}
y_{i}^{φ} = α_{i1}y_{1} + α_{i2}y_{2} + α_{i3}y_{3}
z_{i}^{φ} = α_{i1}z_{1} + α_{i2}z_{2} + α_{i3}z_{3};

and if u = u(x_{1}, …, z_{3}) ∈ R is an arbitrary element, then

u^{φ} = u(x_{1}^{φ}, …, z_{3}^{φ})

is the image of u under the homogeneous linear substitution which replaces x_{1} by x_{1}^{φ}, etc. Clearly this action on R extends to an action of GL(3, F_{0}) on the field of fractions

L_{0} = F_{0}(x_{1}, x_{2}, x_{3}, y_{1}, y_{2}, y_{3}, z_{1}, z_{2}, z_{3}).

So we can regard GL(3, F_{0}) as a group of automorphisms of L_{0}. Let v = x_{1}+y_{2}+z_{3}; and for each g ∈ G, let g̃ = v^{g}. Let Ω_{1} = {g̃ | g ∈ G}. Note that g^{h} = g̃h for all h ∈ G; and if g = (α_{ij}), then

g̃ =α_{11}x_{1} + α_{12}x_{2} + α_{13}x_{3}
+α_{21}y_{1} + α_{22}y_{2} + α_{23}y_{3}
+α_{31}z_{1} + α_{32}z_{2} + α_{33}z_{3}.

Thus the following result is an immediate consequence of Theorem 2.1.

Lemma 2.3. If g̃_{1}, g̃_{2}, h̃_{1}, h̃_{2} ∈ Ω_{1} satisfy the equality

g̃_{1} + 3h̃_{1} = g̃_{2} + 3h̃_{2},

then g̃_{1} = g̃_{2} and h̃_{1} = h̃_{2}.

□

Next let F_{1} be the splitting field over L_{0} of the set of polynomials

P_{0} = {y^{3^{m}} − d | d ∈ L_{0}, m ≥ 1}.

Definition 2.4. Let K be a field of characteristic 0 and let p be a prime. Let t be a transcendental element over K, and let K(t) be the corresponding purely transcendental extension of K. Then K(t, p) denotes the field which is obtained by adjoining elements {t(ℓ) | ℓ ∈ ℕ} to K(t) such that

8 SIMON THOMAS AND BOBAN VELICKOVIC

(a) t(0) = t, and

(b) t(ℓ + 1)^{p} = t(ℓ). Note that the field K(t, p) is uniquely defined up to isomorphism. We shall usually denote the element t(ℓ) by t^{1/p^{ℓ}}.

For the rest of this section, p_{0} = 3, …, p_{4} = 13 will denote the first 5 odd primes. For each C ∈ P̂(G), let Ω_{1}^{C} = Ω_{1}, Ω_{2}^{C} = Ω_{2} = {g̃ + 3ãg | g ∈ G} and Ω_{3}^{C} = Ω_{3} = {g̃+3b̃g | g ∈ G}. Suppose inductively that F_{n} has been defined for some 1 ≤ n ≤ 3. Let t_{n} be a transcendental element over F_{n} and let L_{n} = F_{n}(t_{n}, p_{n}). Then F_{n+1} is the splitting field over L_{n} of the set of polynomials

P_{n} = {y^{2} − (t_{n} − d) | d ∈ Ω_{n}}.

Finally let Ω_{4}^{C} = {c̃ | c ∈ C}. Let t_{4} be a transcendental element over F_{4} and L_{4} = F_{4}(t_{4}, p_{4}). Then F_{C} is the splitting field over L_{4} of the set of polynomials

P_{4}^{C} = {y^{2} − (t_{4} − d) | d ∈ Ω_{4}^{C}}.

Lemma 2.5. If C, D ∈ P̂(G) and there exists g ∈ G such that Cg = D, then F_{C} ≅ F_{D}.

Proof. Regard g as an element of Aut L_{0}. Then it is easily checked that there exists an automorphism π ∈ Aut L_{4} such that π ↾ L_{0} = g and π(t_{n}) = t_{n} for 1 ≤ n ≤ 4. Since c^{g} = c̃g for all c ∈ C, it follows that π maps Ω_{4}^{C} onto Ω_{4}^{D}. Hence F_{C} ≅ F_{D}.

The proof of the converse of Lemma 2.5 is much more involved. Our arguments will closely follow those of Fried and Kollár [FK]. We shall only give the statements of their main two technical lemmas; but in order to make this paper relatively selfcontained, we shall sketch some of their other arguments.

Lemma 2.6 (Fried and Kollár [FK]). Let K be a field of characteristic 0, and let t be a transcendental element over K. Suppose that T_{1}, …, T_{n} are mutually prime nonconstant polynomials in K[t], each having no multiple factors; and for each 1 ≤ i ≤ n, let ϑ_{i} ∈ \overline{K[t]} be an element such that ϑ_{i}^{2} = T_{i}. Let K_{0} = K(t) and for each 1 ≤ i ≤ n, let K_{i} = K(t, ϑ_{1}, …, ϑ_{i}). Then the following statements hold for each 1 ≤ i ≤ n.

FIELDS OF FINITE TRANSCENDENCE DEGREE 9

(a) ϑ_{i} ∉ K_{i−1}.

(b) If η ∈ K_{i} satisfies η^{2} ∈ K_{0}, then there exist an element c ∈ K_{0} and a subset Z of {1, …, i} such that η = c ∏_{ℓ∈Z} ϑ_{ℓ}.

(c) If η ∈ K_{i} and η is algebraic over K, then η ∈ K.

□

Corollary 2.7. Let C ∈ P̂(G) and η ∈ F_{C}. If 1 ≤ n ≤ 4 and η is algebraic over F_{n}, then η ∈ F_{n}.

Proof. We shall prove that for each 1 ≤ n ≤ 3, if η ∈ F_{n+1} is algebraic over F_{n}, then η ∈ F_{n}. The same argument will show that if η ∈ F_{C} is algebraic over F_{4}, then η ∈ F_{4}. The result then follows easily.

Fix some 1 ≤ n ≤ 3. Let p = p_{n} and K = F_{n}. Suppose that η ∈ F_{n+1} is algebraic over F_{n}. Then there exists a finite subset {d_{i} | 1 ≤ i ≤ s} of Ω_{n} such that η is in the splitting field over L_{n} of the set of polynomials

{y^{2} − (t_{n} − d_{i}) | 1 ≤ i ≤ s}.

For each 1 ≤ i ≤ s, let ϑ_{i} ∈ F_{n+1} be an element such that ϑ_{i}^{2} = t_{n} − d_{i}. Choose an integer k > ℓ and an element t ∈ L_{n} such that

(a) t^{p^{k}} = t_{n}, and

(b) η ∈ K(t, ϑ_{1}, …, ϑ_{s}). For each 1 ≤ i ≤ s, let T_{i} = t^{p^{k}} − d_{i} ∈ K[t]. Then the polynomials T_{1}, …, T_{s} satisfy the hypotheses of Lemma 2.6. So applying Lemma 2.6(c), we obtain that η ∈ F = F_{n}.

Definition 2.8. Let K be a field and let p be a prime. Then u ∈ K is said to be a p-high element of K if the equation y^{p^{m}} = u is solvable in K for all m ∈ ℕ.

Lemma 2.9 (Fried and Kollár [FK]). Let K be a field of characteristic 0, and let t be a transcendental element over K. Let p be an odd prime, and let {t(ℓ) | ℓ ∈ ℕ} be a set of elements of \overline{K[t]} such that t(0) = t and t(ℓ + 1)^{p} = t(ℓ). Let

L = K(t(0), t(1), …, t(ℓ), …) = K(t, p).

10 SIMON THOMAS AND BOBAN VELICKOVIC

Let {T_{i} | i ∈ I} be a set of mutually prime polynomials in K[t], none of which is divisible by t or has a multiple factor; and for each i ∈ I, let ϑ_{i} ∈ \overline{K[t]} be an element such that ϑ_{i}^{2} = T_{i}. Let M = L(…, ϑ_{i}, …).

(a) If u ∈ M is a p-high element of M, then u = ct(ℓ)^{m}, for some ℓ ∈ ℕ, m ∈ ℤ and some p-high element c of K.

(b) If p′ is an odd prime such that p′ ≠ p and u ∈ M is a p′-high element of M, then u ∈ K and u is a p′-high element of K.

□

Lemma 2.10. Suppose that p ∈ {p_{n} | 0 ≤ n ≤ 4} and that d is a p-high element of F_{C} for some C ∈ P̂(G).

(a) If p = 3, then d ∈ F_{1}.

(b) If p ≠ 3, then d = et_{n}^{m/p^{ℓ}} for some ℓ ∈ ℕ, m ∈ ℤ and some p-high element e of F_{1}.

Proof. This follows easily from Lemma 2.9.

The following result is a slight variant of Lemma 4.5 of [FK].

Lemma 2.11. Let C ∈ P̂(G) and let 1 ≤ n ≤ 4. Suppose that e ∈ F_{1} ∖ {0}, d ∈ F_{n} ∖ {0} and r = m/p_{n}^{ℓ} for some m ∈ ℤ ∖ {0} and ℓ ∈ ℕ. If the equation y^{2} = e(t_{n}^{r} − d) has a solution in F_{C}, then r = 1, d ∈ Ω_{n}^{C} and e is a square in F_{C}.

Proof. To simplify notation, we shall just consider the case when 1 ≤ n ≤ 3. Let p = p_{n} and K = F_{n}. Let η ∈ F_{C} be a solution of the equation y^{2} = e(t_{n}^{r} − d). By Corollary 2.7, η ∈ F_{n+1}. Thus there exists a finite subset {d_{i} | 1 ≤ i ≤ s} of Ω_{n} such that η is in the splitting field over L_{n} of the set of polynomials

{y^{2} − (t_{n} − d_{i}) | 1 ≤ i ≤ s}.

For each 1 ≤ i ≤ s, let ϑ_{i} ∈ F_{n+1} be an element such that ϑ_{i}^{2} = t_{n} − d_{i}. Choose an integer k > ℓ and an element t ∈ L_{n} such that

(a) t^{p^{k}} = t_{n}, and

(b) η ∈ K(t, ϑ_{1}, …, ϑ_{s}).

FIELDS OF FINITE TRANSCENDENCE DEGREE 11

For each 1 ≤ i ≤ s, let T_{i} = t^{p^{k}} −d_{i} ∈ K[t]. Then the polynomials T_{1}, …, T_{s} satisfy the hypotheses of Lemma 2.6. Note that

η^{2} = e(t_{n}^{r} − d) = e(t^{mp^{k−ℓ}} − d) ∈ K(t).

By Lemma 2.6, there exist polynomials f(t), g(t) ∈ K[t] and a subset Z of {1, …, s} such that

η = \frac{f(t)}{g(t)} ∏_{i∈Z} ϑ_{i}.

It follows that

g^{2}(t)e(t^{mp^{k−ℓ}} − d) = f^{2}(t) ∏_{i∈Z}(t^{p^{k}} − d_{i}). (2.11)

First suppose that m > 0. Then we must have that eg^{2}(t) = f^{2}(t), since the other factors do not have multiple roots. Hence

(t^{mp^{k−ℓ}} − d) = ∏_{i∈Z}(t^{p^{k}} − d_{i}).

This implies that |Z| = 1, and so t^{mp^{k−ℓ}} − d = t^{p^{k}} − d_{i} for some 1 ≤ i ≤ s. Consequently, r = m/p^{ℓ} = 1 and d = d_{i} ∈ Ω_{n}. Since e = f^{2}(t)/g^{2}(t), we also see that e is a square in F_{C}.

Now suppose that m < 0. Then multiplying equation (2.11) by t^{−mp^{k−ℓ}}, we obtain that

g^{2}(t)e(1 − dt^{−mp^{k−ℓ}}) = f^{2}(t) ∏_{i∈Z}(t^{p^{k}} − d_{i})t^{−mp^{k−ℓ}},

and so

−edg^{2}(t)(t^{−mp^{k−ℓ}} − d^{−1}) = f^{2}(t) ∏_{i∈Z}(t^{p^{k}} − d_{i})t^{−mp^{k−ℓ}}.

Arguing as above, we must have that

−edg^{2}(t) = f^{2}(t)t^{2a}

for some integer a such that 0 ≤ −mp^{k−ℓ} − 2a ≤ 1; and

t^{−mp^{k−ℓ}} − d^{−1} = t^{b} ∏_{i∈Z}(t^{p^{k}} − d_{i}),

where b = −mp^{k−ℓ} − 2a. It follows that b = 0 and that −mp^{k−ℓ} = p^{k}. But then p^{k} = 2a, which is impossible since p is odd.

We are now ready to prove the converse of Lemma 2.5.

Theorem 2.12. If C, D ∈ P̂(G) and F_{C} ≅ F_{D}, then there exists g ∈ G such that Cg = D.

Proof. During this proof, isomorphisms will act on the left. (This will not cause any confusion, since we will never need to compose two isomorphisms.) Suppose that π : F_{C} → F_{D} is an isomorphism. If s ∈ F_{1}, then π(s) is a 3-high element of F_{D}; and so Lemma 2.10(a) yields that π(s) ∈ F_{1}. Similarly if 1 ≤ n ≤ 3, then π(t_{n}) is a p_{n}-high element of F_{D}, and so π(t_{n}) ∈ F_{n+1}. Applying Corollary 2.7, we obtain inductively that π [F_{n}] ⊆ F_{n} for all 1 ≤ n ≤ 4. By also considering the isomorphism π^{−1}, we find that π [F_{n}] = F_{n} for all 1 ≤ n ≤ 4.

By Lemma 2.10(b), we have that π(t_{1}) = et_{1}^{r}, where e is a 5-high element of F_{1} and r = m/5^{ℓ} for some m ∈ ℤ ∖ {0} and ℓ ∈ ℕ. Let g̃ ∈ Ω_{1}. Then π(g̃) ∈ F_{1} and the equation

y^{2} = et_{1}^{r} − π(g̃) = e(t_{1}^{r} − e^{−1}π(g̃))

has a solution in F_{D}. So Lemma 2.11 yields that r = 1, e^{−1}π(g̃) ∈ Ω_{1} and e is a square in F_{D}. In particular, there exists an injection φ : Ω_{1} → Ω_{1} such that π(g̃) = eφ(g̃) for all g̃ ∈ Ω_{1}.

Claim 2.13. φ is a bijection.

Proof of Claim 2.13. Let g̃ ∈ Ω_{1}. Since e is a square in F_{D}, the equation

y^{2} = e(t_{1} − g̃) = et_{1} − eg̃

has a solution in F_{D}. Applying π^{−1}, we find that the equation y^{2} = t_{1} − π^{−1}(eg̃) has a solution in F_{C}. By Lemma 2.11, there exists h̃ ∈ Ω_{1} such that π^{−1}(eg̃) = h̃. It follows that φ(h̃) = g̃.

We shall prove that there exists an element g ∈ G such that φ(h̃) = h̃g for all h̃ ∈ Ω_{1}. To see this, we shall next consider π[Ω_{2}]. By Lemma 2.10(b), π(t_{2}) = ft_{2}^{r}, where e is a 7-high element of F_{1} and r = m/7^{ℓ} for some m ∈ ℤ ∖ {0} and ℓ ∈ ℕ. Arguing as above, we find that r = 1 and that f^{−1}π(w) ∈ Ω_{2} for every w ∈ Ω_{2}.

Claim 2.14. f = e.

FIELDS OF FINITE TRANSCENDENCE DEGREE 13

Proof of Claim 2.14. Let w = c̃+ 3ãc ∈ Ω_{2}. Then

f^{−1}π(w) = f^{−1}e (φ(c̃) + 3φ(ãc)) ∈ Ω_{2}.

Thus there exists d ∈ G such that

f^{−1}e = \frac{φ(c̃) + 3φ(ãc)}{d̃+ 3ãd};

and for each h ∈ G,

(\frac{φ(c̃) + 3φ(ãc)}{d̃+ 3ãd}) (φ(h̃) + 3φ(ãh)) ∈ Ω_{2} ⊆ F_{0}[x_{1}, …, z_{3}].

This is possible only if the linear polynomials φ(c̃) + 3φ(ãc) and d̃+ 3ãd are equal. Hence f^{−1}e = 1.

In particular, for all c ∈ G, there exists d ∈ G such that

φ(c̃) + 3φ(ãc) = d̃+ 3ãd.

By Lemma 2.3, it follows that φ(c̃) = d̃ and φ(ãc) = ãd. Hence if we define Ẽ_{a} = {(h̃, ãh) | h ∈ G}, then φ[Ẽ_{a}] ⊆ Ẽ_{a}. Arguing as in the proof of Claim 2.13, we obtain that φ[Ẽ_{a}] = Ẽ_{a}. A similar argument shows that φ also preserves the relation Ẽ_{b} = {(h̃, b̃h) | h ∈ G}. By Lemma 2.2, there exists an element g ∈ G such that φ(h̃) = h̃g for all h̃ ∈ Ω_{1}.

Finally, arguing as above, we find that φ[Ω_{4}^{C}] = Ω_{4}^{D}. (The analogue of Claim 2.14 is the only point in the proof where we make use of the assumption that |C|, |D| ≥ 2.) This implies that Cg = D.

This completes the proof of our main result.

Theorem 2.15. Let σ be the theory of fields of finite transcendence degree. Then (Mod(σ), ≅) is a universal essentially countable Borel equivalence relation.

□

In the remainder of this section, we shall prove that the theory of fields of finite transcendence degree does not admit canonical models. We shall make use of the following results of Hjorth and Kechris [HK].

Theorem 2.16. If σ ∈ L_{ω_{1},ω}, then the following are equivalent.

(i) σ admits canonical models.

(ii) The logic action of S_{∞} on Mod(σ) has the cocycle property.

□

Theorem 2.17. The shift action of the free group G = ⟨a, b⟩ on P(F) does not have the cocycle property.

□

It follows easily that the shift action of G on P̂(G) also does not have the cocycle property.

Theorem 2.18. Let σ be the theory of fields of finite transcendence degree. Then the logic action of S_{∞} on Mod(σ) does not have the cocycle property.

Proof. We shall use the notation which was introduced in the proof of Theorem 2.12, except that we shall once again let all isomorphisms act on the right. Suppose that the Borel cocycle α witnesses that the logic action of S_{∞} on Mod(σ) has the cocycle property. For each C ∈ P̂(G), there exists an element M_{C} ∈ Mod(σ) and a canonical isomorphism π_{C} : F_{C} → M_{C}. Thus whenever F_{C} ≅ F_{D}, then the map

β(F_{C}, F_{D}) = π_{C}α(M_{C}, M_{D})π_{D}^{−1}

is an isomorphism from F_{C} onto F_{D}. Furthermore, if F_{C} ≅ F_{D} ≅ F_{E}, then

β(F_{C}, F_{E}) = β(F_{C}, F_{D})β(F_{D}, F_{E}).

Now define γ as follows. If C, D ∈ P̂(G), let e ∈ F_{1} be such that t_{1}^{β(F_{C},F_{D})} = et_{1}. Then γ(C, D) = g ∈ G is the element such that

h̃^{β(F_{C},F_{D})} = eh̃g

for all h̃ ∈ Ω_{1}. The proof of Theorem 2.12 shows that Cg = D.

Now suppose that F_{C} ≅ F_{D} ≅ F_{E}. Let f ∈ F_{1} be such that t_{1}^{β(F_{D},F_{E})} = ft_{1} and let γ(D, E) = k. Note that

t_{1}^{β(F_{C},F_{E})} = t_{1}^{β(F_{C},F_{D})β(F_{D},F_{E})}
= (et_{1})^{β(F_{D},F_{E})}
= e^{β(F_{D},F_{E})}ft_{1}.

Hence for every h̃ ∈ Ω_{1},

h̃^{β(F_{D},F_{E})} = h̃^{β(F_{C},F_{D})β(F_{D},F_{E})}
= (eh̃g)^{β(F_{D},F_{E})}
= e^{β(F_{D},F_{E})}fh̃gk.

It follows that γ(C, E)γ(C, D) = γ(D, E). But then γ witnesses that the action of G on P̂(G) has the cocycle property, which is a contradiction.

Corollary 2.19. The theory of fields of finite transcendence degree does not admit canonical models.

□

3. A linear representation of the free group

In this section, we shall prove Theorem 2.1. Let u be a real number which is transcendental over ℚ; and let 0 < θ < π be the real number such that θ = 2 arctan u. Then

sin θ = \frac{2u}{1 + u^{2}} and cos θ = \frac{1 − u^{2}}{1 + u^{2}} .

Let A and B be the real orthogonal matrices defined by

A = (\begin{matrix}cos θ & − sin θ & 0 \\ sin θ & cos θ & 0 \\ 0 & 0 & 1 \end{matrix}) and B = (\begin{matrix}1 & 0 & 0 \\ 0 & cos θ & − sin θ \\ 0 & sin θ & cos θ \end{matrix}) .

In [dG], de Groot proved that ⟨A, B⟩ is a free subgroup of GL(3, ℚ(u)). We shall prove that G satisfies the requirements of Theorem 2.1.

Suppose that C ∈ G is any element. Then each entry c of C has a unique expression of the form

c = α + ∑_{i=1}^{∞} (β_{i} sin θ + γ_{i} cos θ) cos^{i−1} θ,

where α, β_{i}, γ_{i} ∈ ℤ and β_{i} = γ_{i} = 0 for all but finitely many integers i. This expression will be called the normal form of c; and throughout this section, we shall suppose that all of the entries of each C ∈ G have been written in normal form. For example, suppose that n > 0 and that C = A^{n} ∈ G. Then

C = (\begin{matrix}cos nθ & − sin nθ & 0 \\ sin nθ & cos nθ & 0 \\ 0 & 0 & 1 \end{matrix}) .

Using the identities,

cos nθ = 2^{n−1} cos^{n} θ + terms of lower degree in cos θ

and

sin nθ = sin θ (2^{n−1} cos^{n−1} θ + terms of lower degree in cos θ)

we easily obtain the following result.

Lemma 3.1. Suppose that C = A^{n} ∈ G, where n ∈ ℤ ∖ {0}. Then

C = (\begin{matrix}Q cos θ & − sgn(n)P sin θ & 0 \\ sgn(n)P sin θ & Q cos θ & 0 \\ 0 & 0 & 1 \end{matrix})

where P and Q are polynomials in cos θ with integer coefficients of degree d = |n|−1 and leading coefficient 2^{|n|−1}.

□

Similary we obtain the following result.

Lemma 3.2. Suppose that C = B^{m} ∈ G, where m ∈ ℤ ∖ {0}. Then

C = (\begin{matrix}1 & 0 & 0 \\ 0 & Q cos θ & − sgn(m)P sin θ \\ 0 & sgn(m)P sin θ & Q cos θ \end{matrix})

where P and Q are polynomials in cos θ with integer coefficients of degree d = |m|−1 and leading coefficient 2^{|m|−1}.

□

Finally we shall require the following the result from page 59 of Wagon [W].

Lemma 3.3. Suppose that C = A^{n_{1}}B^{m_{1}} … A^{n_{s}}B^{m_{s}} ∈ G, where s ≥ 1 and n_{i}, m_{i} ∈ ℤ ∖ {0} for 1 ≤ i ≤ s. Let Σ = |n_{1}| + |m_{1}| + ⋯ + |n_{s}| + |m_{s}|. Then

C = (\begin{matrix}P_{1} cos θ & − sgn(n_{1})Q_{1} sin θ & − sgn(n_{1}m_{s})Q_{2} cos θ \\ P_{2} sin θ & Q_{3} cos θ & − sgn(m_{s})Q_{4} sin θ \\ P_{3} cos θ & P_{4} sin θ & P_{5} cos θ \end{matrix})

where

(a) each Q_{ℓ} is a polynomial in cos θ with integer coefficients of degree d = Σ − 1 and leading coefficient 2^{Σ−2s}; and

(b) each P_{ℓ} is a polynomial in cos θ with integer coefficients of degree strictly less than d.

□

Suppose now that the matrices C_{1}, C_{2}, D_{1}, D_{2} ∈ G satisfy the equality

C_{1} + 3D_{1} = C_{2} + 3D_{2}.

If C_{1} = C_{2}, then D_{1} = D_{2}. So suppose that C_{1} ≠ C_{2}. After multiplying both sides by C_{2}^{−1} if necessary, we can suppose that C_{2} is the identity matrix I. So we have the equality

C_{1} − I = 3 (D_{2} − D_{1}) .

Also, after conjugating both sides by a suitable element of G if necessary, we can suppose that C_{1} has either the form A^{n} for some n ∈ ℤ ∖ {0}, or B^{m} for some m ∈ ℤ ∖ {0}, or A^{n_{1}}B^{m_{1}} … A^{n_{s}}B^{m_{s}} for some s ≥ 1 and n_{i}, m_{i} ∈ ℤ ∖ {0}. Let C_{1} − I = (c_{ij}). Then applying Lemmas 3.1, 3.2 and 3.3, we see that c_{22} is a polynomial in cos θ with integer coefficients of degree d ≥ 1 and leading coefficient 2^{ℓ} for some ℓ ≥ 0. On the other hand, since (c_{ij}) = 3 (D_{2} − D_{1}), it follows that each coefficient of c_{22} is divisible by 3, which is a contradiction. This completes the proof of Theorem 2.1.

4. Concluding remarks

In this paper, we have made no attempt to keep the transcendence degree of F_{C} as low as possible. (For example, the proof of Theorem 2.1 shows that we could have taken R = F_{0}[y_{1}, y_{2}, y_{3}]; which would have lowered the transcendence degree from 14 to 8.) However, we believe that the following question is of some interest.

Question 4.1. Suppose that φ is the theory of fields of transcendence degree 1. Is (Mod(φ), ≅) a universal essentially countable Borel equivalence relation?

References

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[DJK] R. Dougherty, S. Jackson and A. Kechris, The structure of hyperfinite Borel equivalence relations, Trans. Amer. Math. Soc. 341 (1994), 193–225.

[dG] J. de Groot, Orthogonal isomorphic representations of free groups, Can. J. Math 8 (1956), 256–262.

[FK] E. Fried and J. Kollár, Automorphism groups of fields, in Universal Algebra (E. T. Schmidt et al., eds.), Coloq. Math. Soc. Janos Boyali, vol 24, 1981, pp. 293–304.

[FS] H. Friedman and L. Stanley, A Borel reducibility theory for classes of countable structures, Journal of Symbolic Logic 54 (1989), 894–914.

[HK] G. Hjorth and A. Kechris, Borel equivalence relations and classification of countable models, Annals of Pure and Applied Logic 82 (1996), 221–272.

[JKL] S. Jackson, A. Kechris, and A. Louveau, Countable Borel equivalence relations, in preparation.

[Ke1] A. Kechris, Countable sections for locally compact group actions, Ergodic Theory Dynamical Systems 12 (1992), 283–295.

[Ke2] A. Kechris, Classical Descriptive Set Theory, Springer-Verlag, 1995.

[TV] S. Thomas and B. Velickovic, On the complexity of the isomorphism relation for finitely generated groups, preprint (1998).

[W] S. Wagon, The Banach-Tarski Paradox, Encyclopedia of Mathematics and its Applications 24, Cambridge University Press, 1985.

Mathematics Department, Rutgers University, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019, USA

Equipe de Logique, URA 753, Universite de Paris 7, 2 Place Jussieu, France