Amalgamation of types in pseudo-algebraically closed fields and applications
aa r X i v : . [ m a t h . L O ] D ec Amalgamation of types in pseudo-algebraically closedfields and applications
Zo´e Chatzidakis ∗ DMA (UMR 8553), Ecole Normale Sup´erieureCNRS, PSL Research UniversityDecember 27, 2018
Abstract
This paper studies unbounded PAC fields and shows an amalgamation result for typesover algebraically closed sets. It discusses various applications, for instance that omega-free PAC fields have the property NSOP3. It also contains a description of imaginaries inPAC fields.
Introduction
Pseudo-algebraically closed fields (henceforth abbreviated by PAC) were introduced by Ax inhis famous paper [1] on the theory of finite fields. The elementary theory of arbitrary PACfields, studied among others by Cherlin-Van den Dries-Macintyre [6] and by Ershov [10], puts inlight an interesting dichotomy: definable sets are given, on the one hand by classical algebraicdata, and on the other hand by elementary statements concerning the Galois group. Many ofthe properties of the theory of a PAC field thus reduce to the corresponding properties of itsGalois group. For instance, if the subfield of algebraic numbers of the PAC field F is decidable,then Th( F ) will be decidable if and only if the “theory” of its absolute Galois group is decidable.One also knows that the structure of their models is complicated: a result of Duret ([9]) assertsthat a PAC field which is not separably closed has the independence property.Interest for the model theory of PAC fields revived in the mid 90’s, when Hrushovski andPillay ([14]) were able to use stability theoretic techniques for groups definable in pseudo-finitefields, and more generally in bounded PAC fields (a field is bounded if for each n > n ). It was then observed that bounded PAC fieldshave a simple theory, because they satisfy the independence theorem (1991 result of Hrushovski,only published in 2005, [13]). Other results with a stability-theoretic flavour followed: in [3], ∗ Most of this work was done while the author was partially supported by MRTN-CT-2004-512234 and byANR-06-BLAN-0183, while a member of the Equipe de Logique Math´ematiques (UMR 7056), in UniversityParis Diderot. The work achieved its final form in 2017, while the author was partially supported by ANR-13-BS01-0006. ω -free PAC fields.The author shows that for these fields, forking is the transitive closure of weak independence,and shows versions of the independence theorem for various independence notions, the mostdifficult one being that ω -free PAC fields of chararacteristic 0 satisfy the independence theoremwith independence being the genuine non-forking. This last result is quite surprising, giventhat the theories of ω -free PAC fields are not simple. This suggested that more can be doneon unbounded PAC fields, and that their study might provide an insight of good behaviours ofmodels of non-simple theories.In this paper we continue the investigation of the behaviour of unbounded PAC fields. Ourmain result is an amalgamation result for types, similar to the (weak) independence theoremof [4]. This result (Theorem 2.1) isolates the conditions under which amalgamation of types ispossible. It has various consequences, notably a weak independence theorem over models forPAC fields F such that S G ( F ) has a simple theory (Theorem 2.5), and the fact that Frobeniusfields satisfy NSOP (see 2.9 and 2.10). It also appears as an ingredient in the description ofimaginaries in PAC fields of finite degree of imperfection: an imaginary of the PAC field F isequi-definable with a finite collection of pairs ( a, D ), where a is a tuple of elements of F and D is an imaginary of S G ( F ) (Theorem 4.2). We show by an example that this result is bestpossible.The hope that PAC fields might provide good examples of things happening beyond sim-plicity was vindicated. Recent results of Chernikov and Ramsey ([7], Theorem 6.2) show thatthe weak independence theorem proved in [4] for Frobenius fields implies that the theory of aFrobenius field is NSOP . Thus these fields provide a large family of new examples of struc-tures with an NSOP theory. This is particularly useful as very few examples of theories withNSOP were known. It can be hoped that a further study of these PAC fields might lead tonew insight on NSOP theories. The ω -free PAC fields are particularly nice Frobenius fields, inwhich types and definable sets are well understood. As we show here, imaginaries are equallywell understood.Clearly, the connections between the neo-stability properties of the Galois group of a PACfield and those of the field also need to be explored further. Results of Nick Ramsey ([18])suggest this is the case for the properties NSOP and NTP .The paper is organised as follows. In section 1, after setting up the notation, we recall orprove some technical results on fields and profinite groups. Section 2 contains the main resultof this paper, Theorem 2.1, as well as various independence theorems and SOP n properties for n ≥
3. We conclude section 2 with some questions. Section 3 develops the part of the logic ofcomplete systems which is interpretable in fields. In particular, it sets up the formalism whichwill enable us to deal with definable sets. This is applied in section 4 (Theorem 4.2) to give thedescription of imaginaries of PAC fields F of finite degree of imperfection.2 Notation and preliminary results
Recall first that a field F is PAC if every absolutely irreducible variety defined over F has an F -rational point. Equivalently, if F is existentially closed in any regular extension. In this sectionwe set up the notation, recall some classical results on PAC fields, and give two additionallemmas. We assume familiarity with elementary results on field extensions, see e.g. ChapterIII of [16]. . We work in the usual language of rings ( { + , − , · , , } ), some-times expanded by adding constants for a p -basis. The separable closure of a field K is denotedby K s , and its absolute Galois group G al ( K s /K ) by G ( K ). If A ⊆ K , then acl( A ) denotes themodel-theoretic closure of A in the sense of Th( K ). It is known that K is a regular extensionof acl( A ).We will often work inside the separable closure of a field K . In that case, we will denote bySCF the theory Th( K s ), the notation tp SCF ( ) will refer to the type in the field K s . We usethe notation acl K s ( A ) to denote the algebraic closure in the sense of Th( K s ), i.e., the smallestsubfield of K s containing A and of which K s is a regular extension. We will say that twosubsets of K (or of K s ) are SCF-independent over some E if they are independent in the senseof Th( K s ).In addition, unless otherwise specified, all fields will be subfields of some large algebraicallyclosed field Ω. If A, B are two subfields, then AB denotes the composite field.An extremely useful and fundamental result on PAC fields is the so-called “embeddinglemma” of Jarden and Kiehne: Theorem 1.2. (Lemma 20.2.2 in [12])
Let
E/L and
F/M be separable field extensions sat-isfying: E is countable and F is an ℵ -saturated PAC field; if char( F ) = p > , assume inaddition that [ E : E p ] ≤ [ F : F p ] . Assume that there is an isomorphism ϕ : L s → M s suchthat ϕ ( L ) = M , and a commutative diagramme G ( E ) Φ ←−−− G ( F ) res y y res G ( L ) Φ ←−−− G ( M ) where Φ : σ ϕ − σϕ , is the dual of ϕ , and Φ is a (continuous) homomorphism. Then ϕ extends to an embedding ϕ : E s → F s , with dual Φ , and such that F/ϕ ( E ) is separable. Remarks 1.3.
We will use the following essentially immediate consequences of this result.(1) We may replace the countability hypothesis on E by asking F to be | E | + -saturated. Theproof is identical.(2) We will usually have that the extensions E/L and
F/M are regular. This means that therestriction maps G ( E ) → G ( L ) and G ( F ) → G ( M ) are onto. Note that the conclusionwill then be that F/ϕ ( L ) is regular. Similarly, if Φ is onto, then the extension F/ϕ ( E )will be regular. 33) (Notation as above.) Let E ′ be a Galois extension of E containing L s , and Φ ′ : G ( F ) →G al ( E ′ /E ) such that the following diagramme commutes: G al ( E ′ /E ) Φ ′ ←−−− G ( F ) res y y res G ( L ) Φ ←−−− G ( M )As G ( F ) is projective, the map Φ ′ factors through a homomorphism Φ : G ( F ) → G ( E ) (seeTheorem 11.6.2 in [12]). Applying the embedding lemma therefore gives us an embedding ϕ ′ : E ′ → F s , with dual Φ ′ . Complete systems associated to profinite groups
Cherlin, Van den Dries and Macintyre show in [6] how to associate to any profinite group G a structure SG in an ω -sorted language L G , called the complete system of G , which encodesprecisely the inverse system of all finite continuous quotients of G . The functor G SG isa contravariant functor, and defines a duality between the category of profinite groups withcontinuous epimorphisms and the category of complete systems with embeddings. The functordual to S is the functor G which to a complete system S associates the inverse limit of theinverse system of finite groups given by S . An important remark, which is at the core ofthe results of Cherlin Van den Dries and Macintyre, is that the functor S G commutes withultraproducts and therefore with ultrapowers: If U is an ultrafilter on a set I and K is a field,then S G ( K U ) ≃ ( S G ( K )) U , where the second ultraproduct is taken in the ω -sorted context(i.e., sort by sort). Hence, K ≡ L implies S G ( K ) ≡ S G ( L ). In an unpublished manuscript,Cherlin, Van den Dries and Macintyre also show that this ω -sorted logic on S G ( K ) is in somesense the strongest logic of the Galois group G ( K ) which is interpretable in the field K . Formore details on complete systems and their logic, see [6] or the Appendix of [4].We will first briefly recall the notation and definitions for arbitrary profinite groups, beforegoing to the setting of Galois groups. .Let G be a profinite group, and L G be the ω -sorted language with sorts indexed by the positiveintegers, and with non-logical symbols {≤ , C, P, } , where ≤ and C are binary relations, P isa ternary relation and 1 is a constant symbol. The complete system associated to G is the L G -structure S ( G ), with universe the disjoint union S · N G/N where N ranges over all normalopen subgroups of G . An element of G/N , i.e. a coset gN , will be of sort n if and only if[ G : N ] ≤ n , and 1 = G is the only element of sort 1. We have gN ≤ hM ⇐⇒ N ⊆ M , C ( gN, hM ) ⇐⇒ gN ⊆ hM , and P ( g N , g N , g N ) ⇐⇒ N = N = N and g g N = g N . The class of complete systems of profinite groups is the class of models of a theory T G . The functor S defines a duality between the category of profinite groups with continuousepimorphisms and the category of models of T G with embeddings. .Let F be a field, E a Galois extension of F , and G = G al ( E/F ). The universe of SG is thedisjoint union of all G al ( L/F ) where L is a finite Galois extension of F contained in E . The4lements of sort n with be the Galois groups of size ≤ n . The language L G is interpreted asfollows: G al ( F/F ) = 1; whenever L ⊇ L , C ∩ ( G al ( L /F ) × G al ( L /F )) is the graph of therestriction maps G al ( L /F ) → G al ( L /F ) and ≤ contains G al ( L /F ) × G al ( L /F ); the ternaryrelation P encodes the graph of multiplication on each G al ( L/F ).A subset S of SG is a subsystem of SG if it has the following two properties: (i) ∀ σ, τ ∈ S, ∃ ρ ∈ S ( ρ ≤ σ ∧ ρ ≤ τ ); (ii) If σ ∈ S and τ ≥ σ , then τ ∈ S . If A ⊂ SG , then h A i denotesthe subsystem of SG generated by A .One sees easily that if S is a subsystem of SG , then S = S G al ( M/F ), where M is thecomposite of all Galois extensions L such that S contains G al ( L/F ). The inclusion map S ⊂ SG and the restriction map G al ( E/F ) → G al ( M/F ) are dual of each other .Let F and F be fields, and ϕ : F s → F s an embedding such that ϕ ( F s ) ∩ F = F . Wethen get a continuous epimorphism Φ : G ( F ) → G ( F ), defined by σ ϕ − σϕ (the dual of ϕ ).Applying the functor S to Φ gives us an embedding S G ( F ) → S G ( F ), defined as follows: if L is a finite Galois extension of F and σ ∈ G al ( L /F ), then S Φ( σ ) is the unique element of G al ( F ϕ ( L ) /F ) extending the element ϕσϕ − of G al ( ϕ ( L ) /ϕ ( F )). We call the map S Φ the double dual of ϕ . Theorem 1.6. (Cherlin, Van den Dries, Macintyre [6]). Let F and F be PAC fields, separableover a common subfield E . The following conditions are equivalent:(1) F ≡ E F .(2) (i) F and F have the same degree of imperfection,(ii) There is ϕ ∈ G ( E ) such that ϕ ( F ∩ E s ) = F ∩ E s , and the double dual S Φ : S G ( F ∩ E s ) → S G ( F ∩ E s ) of ϕ , is a partial elementary L G -map S G ( F ) → S G ( F ) .(In particular, S G ( F ) ≡ S G ( F ) ). From this result, one easily deduces a description of types:
Theorem 1.7.
Let F be a PAC field, separable over some subfield E . Let a and b be tuples ofelements of F , and A = acl K s ( E, a ) ∩ F , B = acl K s ( E, b ) ∩ F . The following conditions areequivalent: (1) tp ( a/E ) = tp ( b/E ) . (2) There is an E -isomorphism ϕ : A s → B s , with ϕ ( a ) = b , ϕ ( A ) = B , such that the doubledual S Φ : S G ( A ) → S G ( B ) is a partial elementary L G -map of S G ( F ) . . If F is a PAC field and A ⊂ F , then acl( A ) =acl F s ( A ) ∩ F (see 4.5 in [5]). Let E ⊂ A, B be subfields of F , and assume that A and B are SCF-independent over E . If char ( F ) = p > F : F p ] < ∞ assume moreoverthat E contains a p -basis of F . Then acl F s ( AB ) = (acl F s ( A )acl F s ( B )) s . Hence we also haveacl( AB ) = (acl( A )acl( B )) s ∩ F . Warning: in earlier papers by the author they are called substructures . robenius and ω -free PAC fields Definition 1.9. (1) A profinite group G has the embedding property if for any finite groups A , B , whenever f : G → A and g : B → A , f ′ : G → B are (continuous) epimorphisms,then there exists an epimorphism h : G → B such that f = g ◦ h : G f (cid:15) (cid:15) ∃ h (cid:127) (cid:127) ⑦ ⑦ ⑦ ⑦ B g / / A This property translates into a property of SG which is axiomatisable in the language L G . See section 24.3 of [12] for more details and properties of these groups.(2) A Frobenius field is a PAC field whose absolute Galois group G ( F ) has the embeddingproperty.(3) Recall that a PAC field F is ω -free if whenever F ≺ F is countable, then G ( F ) ≃ ˆ F ω ,the free profinite group on ℵ generators. In particular (or equivalently, using a result ofIwasawa), all finite groups occur as finite quotients of G ( F ), and F is Frobenius.Being Frobenius is an elementary property of a field F . When dualized, and if S G ( F ) iscountable, it says that any L G -isomorphism between two finite subsystems of S G ( F ) extendsto an automorphism of S G ( F ). In particular this implies the following: Theorem 1.10. ([6]) If F is a Frobenius field, then any L G -isomorphism between two subsys-tems of S G ( F ) is elementary. Thus, in Theorems 1.6 and 1.7, the conditions stating that the partial maps S Φ are elementarycan be removed. ω -sorted logic behaves very much like ordinary one-sorted logic, provided one workssort by sort. Our ω -sorted structure S can be viewed as the countable union of structures S n , n ≥
1, where each S n has universe the elements of sort ≤ n , and is a structure in thelanguage with n sorts, relational symbols P, C and ≤ , constant symbol 1. The theory of S isthen naturally the limit of the theories of the S n ’s. For instance, let G be a profinite groupwith the embedding property, SG its complete system. Then the above characterisation ofcountable models of Th( SG ) translates into: Th( SG ) is ℵ -categorical (see [6]). Notions suchas stability or ω -stability easily generalise: one just counts types in each sort. Notions whichare local immediately generalise, as they only involve finitely many sorts. For instance, theusual definition of forking of a formula over a set; and therefore forking of a type: a type willfork over a set if it contains a formula which forks over that set. Hence one can define theproperty of a theory of being simple. Note that Th( S ) is simple if and only if Th( S n ) is simplefor every n ≥
1. We will use the fact that the results of Kim and Pillay characterizing simpletheories via the properties of the forking relation go through.Before proving our two technical lemmas, we first recall some results from [4]:6 emma 1.12. (Lemma 2.1 in [4])
Let A , B be fields (contained in Ω ), and assume that thefield composite AB is a regular extension of A and of B . If E = A ∩ B , then E s = A s ∩ B s . Lemma 1.13.
Let A , B , C , E be separably closed fields ( ⊂ Ω ), with A, B, C separable ex-tensions of E , and AB a separable extension of A and of B , free from C over E , and with A ∩ B = E . Then(i) (a) ( AB ) s ∩ ( AC ) s ( BC ) s = AB ; (b) ( AB ) s C ∩ ( AC ) s ( BC ) s = ABC (ii) (a) ( AC ) s ∩ ( AB ) s ( BC ) s = AC ; (b) ( AC ) s B ∩ ( AB ) s ( BC ) s = ABC .Proof . Items (4) and (2) of Lemma 2.5 in [4] give (i)(a)(b) and (ii)(a). By (3) of that samelemma, we have ( AC ) s ( AB ) s ∩ ( BC ) s ( AB ) s = C ( AB ) s , which implies ( AC ) s B ∩ ( AB ) s ( BC ) s ⊆ C ( AB ) s ∩ ( AC ) s B ⊆ ABC by (i)(b), and gives us (ii)(b).
Lemma 1.14.
Let A , B , C (contained in Ω ) be regular extensions of a field E , and assumethat AB is a regular extension of A and of B , that A ∩ B = E , and that AB is free from C over E . Consider the map ρ : G al (( AB ) s ( AC ) s ( BC ) s /ABC ) → G ( AB ) × G ( AC ) × G ( BC ) defined by σ ( σ | ( AB ) s , σ | ( AC ) s , σ | ( BC ) s ) . Then the image of ρ is the subgroup of G ( AB ) × G ( AC ) × G ( BC ) consisting of the triples ( σ , σ , σ ) such that σ | A s = σ | A s , σ | B s = σ | B s , σ | C s = σ | C s . Proof . The compatibility conditions are clearly necessary, it remains to show that they aresufficient. Let ( σ , σ , σ ) ∈ G ( AB ) × G ( AC ) × G ( BC ) satisfy the required conditions. We willfirst show that there is some σ ∈ G al ( A s B s C s /ABC ) which agrees with σ on A s B s , with σ on A s C s and with σ on B s C s . First note that σ , σ and σ all agree on E s .As C is free from AB over E , and is a regular extension of E , we know that C s is lin-early disjoint from ( AB ) s over E s , and therefore from A s B s C over CE s . Hence there is σ ∈ G al ( A s B s C s /ABC ) which agrees with σ on A s B s and with σ on C s ; by hypothesis, σ therefore agrees with σ on A s C s , and with σ on B s and on C s , i.e., on B s C s .By Lemma 1.12, A s ∩ B s = E s and we may apply Lemma 1.13 to obtain: • the Galois extensions ( AB ) s C s , ( AC ) s B s and A s ( BC ) s are linearly disjoint over A s B s C s (use (i)(b) and (ii)(b)), whence G al (( AB ) s ( AC ) s ( BC ) s /A s B s C s ) ≃G al (( AB ) s C s /A s B s C s ) × G al (( AC ) s B s /A s B s C s ) × G al ( A s ( BC ) s /A s B s C s ); • A s B s C s is a regular extension of A s B s , because A s B s C s ∩ ( AB ) s = A s B s (by (i)(a)), andtherefore G al (( AB ) s C s /A s B s C s ) ≃ G ( A s B s );7 A s B s C s is a regular extension of A s C s , because A s B s C s ∩ ( AC ) s = A s C s (by (ii)(a)), sothat G al (( AC ) s B s /A s B s C s ) ≃ G ( A s C s ); • A s B s C s is a regular extension of B s C s (as above using (ii)(a)), so that G al (( BC ) s A s /A s B s C s ) ≃ G ( B s C s ) . This gives in particular that G al (( AB ) s ( AC ) s ( BC ) s /A s B s C s ) ≃ G ( A s B s ) × G ( A s C s ) × G ( B s C s ) . Hence, the automorphism σ ∈ G al ( A s B s C s /ABC ) can be lifted uniquely to an element σ of G al (( AB ) s ( AC ) s ( BC ) s /ABC ) which agrees with σ on ( AB ) s , with σ on ( AC ) s and with σ on ( BC ) s . Lemma 1.15.
Let E ⊂ A and E be algebraically closed subsets of a PAC field F , and assumethat ϕ : E s → E s is an isomorphism and restricts to an elementary map E → E . If thecharacteristic is p > and [ F : F p ] < ∞ , then assume that E contains a p -basis of F . Let S Ψ : S G ( A ) → S ⊂ S G ( F ) be a partial L G -elementary isomorphism extending the double dual S Φ : S G ( E ) → S G ( E ) of ϕ . Then in some elementary extension F ∗ of F , there is B ,which is SCF-independent from F over E , and an isomorphism ϕ : A s → B s sending A to B ,extending ϕ and with double dual S Ψ . Moreover, ( B, E ) realises tp ( A, E ) .Proof . Let M be the Galois extension of F with Galois group over F corresponding to S , i.e.,the restriction map res : G ( F ) → G al ( M/F ) is dual to S ⊂ S G ( F ). (Without the requirementthat B be SCF-independent from F over E , we could just apply Theorem 1.2.)Choose any extension ϕ of ϕ to A s such that ϕ ( A s ) and F s are linearly disjoint over E s ,and let B = ϕ ( A ). Then the double dual S Φ of ϕ extends S Φ , and the dual Φ of ϕ definesan isomorphism G ( B ) → G ( A ) which induces the dual Φ of ϕ , Φ : G ( E ) → G ( E ). Thedual Ψ of S Ψ defines an isomorphism Ψ : G al ( M/F ) → G ( A ), which also induces Φ on G ( E ).Consider the profinite group H = { (Φ − ( σ ) , Ψ − ( σ )) | σ ∈ G ( A ) } ⊆ G ( B ) × G al ( M/F ) . Then H is the graph of Ψ − Φ : G ( B ) → G al ( M/F ), and can be identified with a closedsubgroup of G al ( B s M/BF ) ≃ G ( B ) × G ( E ) G al ( M/F ). Let L be the subfield of B s M fixedby the elements of H . Since H projects onto G ( B ) and onto G ( S ) = G al ( M/F ), it followsthat L is a regular extension of B and of F , and that G al ( B s M/L ) = H canonically identifieswith G ( B ) and with G al ( M/F ) = G ( S ) via the restriction maps. It follows that the restriction G al ( B s F s /L ) → G ( F ) is an isomorphism. By Theorem 1.2, there is an elementary extension F ∗ of F containing B , and such that F ∗ ∩ B s F s = L . Then the map ϕ : A s → B s is ourdesired map: by construction, inside S G ( F ∗ ), we have S G ( B ) = S , and the double dual of ϕ coincides with S Ψ, which is an elementary map. This proves the first assertion, and 1.7 givesthe moreover part. 8 emark 1.16.
Let
E, E , A, ϕ be as above. Let L be a Galois extension of A , and assumethat we have a partial elementary L G -map S Ψ ′ : S G al ( L/A ) → S ′ ⊂ S G ( F ) which extends therestriction of S Φ to S G al ( L ∩ E s /E ). Then there is an elementary extension F ∗ of F suchthat the map S Ψ ′ extends to an L G -embedding S G ( A ) → S G ( F ∗ ). Thus in the above lemma,we may replace S G ( A ) by a subsystem. Remark 1.17.
Notation as in 1.15. The additional hypothesis in Lemma 1.15 when p > E : E p ] < ∞ is actually not necessary but the proof needs to be slightly modified. Indeed, theproblem occurs if [ E : E p ] < [ A : A p ] since then one will get [ L : L p ] > [ F : F p ] and we cannotapply the embedding lemma. This is not hard to fix: one selects a p -basis S of B over E (andtherefore of L over F ), and forces it to realise the generic | S | -type of Th( F s ) over F s . Detailsare left to the reader. Theorem 2.1.
Let F be a PAC field, and let E, A, B, C , C be algebraically closed subsets of F , with E contained in A, B, C , C . Assume that A ∩ B = E , that A and C , and B and C , are SCF-independent over E , and that if the degree of imperfection of F is finite, then E contains a p -basis of F . Moreover, assume that there is an E s -isomorphism ϕ : C s → C s suchthat ϕ ( C ) = C , and that there is S ⊂ S G ( F ) , and elementary (in S G ( F ) ) isomorphisms S Ψ : h S G ( C ) , S G ( A ) i → h S , S G ( A ) i S Ψ : h S G ( C ) , S G ( B ) i → h S , S G ( B ) i such that(i) S Ψ is the identity on S G ( A ) , S Ψ is the identity on S G ( B ) , S Ψ i ( S G ( C i )) = S and(ii) if S Φ : S G ( C ) → S G ( C ) is the morphism double dual to ϕ , then S Ψ S Φ = S Ψ | S G ( C ) . Then, in some elementary extension F ∗ of F , there is C which is SCF-independent from ( A, B ) over E , realises tp ( C /A ) ∪ tp ( C /B ) , and with S G ( C ) = S (The variables for tp ( C /A ) and tp ( C /B ) are identified via ϕ .)Proof . We may assume that F is sufficiently saturated; then S G ( F ) will also be sufficientlysaturated. We work inside Ω. Choose C realising tp SCF ( C /E ), and SCF-independent from F over E . Let ϕ : C s → C s and ϕ : C s → C s be E s -isomorphisms such that ϕ ( C ) = C , and ϕϕ = ϕ , (whence ϕ ( C ) = C ) . As A is linearly disjoint from C and from C over E , we have that A s is linearly disjoint from C s and from C s over E s , and we may therefore extend ϕ to an A s -isomorphism ϕ ′ : ( AC ) s → ( AC ) s . ϕ to a B s -isomorphism ϕ ′ : ( BC ) s → ( BC ) s . Let D = ϕ ′ − (acl( AC )), and D = ϕ ′ − (acl( BC )). Then D ⊆ ( AC ) s , D ⊆ ( BC ) s .Because S Ψ and S Ψ are elementary and S G ( F ) is sufficiently saturated, there are subsys-tems S and S of S G ( F ), and elementary isomorphisms S Ψ ′ : S G (acl( AC )) → S , S Ψ ′ : S G (acl( BC )) → S extending S Ψ and S Ψ respectively.For i = 0 , , L i be the Galois extension of F such that the restriction map G ( F ) →G al ( L i /F ) is dual to S i ⊂ S G ( F ). Let S Φ ′ i be the double dual of ϕ ′ i for i = 1 ,
2, and define S Θ i = S Ψ ′ i S Φ ′ i : S G ( D i ) → S i , and let Θ i : G al ( L i /F ) → G ( D i )be the homeomorphism dual to S Θ i . We will show that there is a continuous morphism (notnecessarily onto) Θ : G al (( AB ) s L L /F ) → G al (( AB ) s ( AC ) s ( BC ) s /ABC )which induces Θ i on G al ( L i /F ) for i = 1 ,
2, the identity on G (acl( AB )), and whose image U projects onto G ( D ), G ( D ) and G (acl( AB )) (via the restriction maps).Since A = acl( A ) and B = acl( B ), we know that F is a regular extension of A andof B ; hence AB is a regular extension of A and of B . By Lemma 1.14, we may identify G al (( AB ) s ( AC ) s ( BC ) s /ABC ) with the set of triples ( σ , σ , σ ) ∈ G ( AB ) × G ( AC ) × G ( BC )satisfying σ | A s = σ | A s , σ | B s = σ | B s , σ | C s = σ | C s . For σ ∈ G al (( AB ) s L L /F ) we defineΘ( σ ) = ( σ | ( AB ) s , Θ ( σ | L ) , Θ ( σ | L )) =: ( σ , σ , σ ) . We need to show that Θ( σ ) ∈ G al (( AB ) s ( AC ) s ( BC ) s /ABC ). Because ϕ ′ is an A s -isomorphism, S Φ ′ is the identity on S G ( A ), and because S Ψ ′ extends S Ψ , so is S Ψ ′ . Hence S Θ is theidentity on S G ( A ). This shows that σ | A s = σ | A s . Similarly, σ | B s = σ | B s . We still need toshow that σ | C s = σ | C s . By duality, it is enough to show that S Θ and S Θ agree on S G ( C ).We know that ϕϕ = ϕ , and that ϕ ′ i extends ϕ i . Hence S Φ S Φ ′ | S G ( C ) = S Φ ′ | S G ( C ) . We also have that S Ψ | S G ( C ) = S Ψ S Φ . Hence S Θ | S G ( C ) = S Ψ | S G ( C ) S Φ ′ | S G ( C )
10 ( S Ψ S Φ)( S Φ − S Φ ′ | S G ( C ) )= S Θ | S G ( C ) , so that Θ takes its values in G al (( AB ) s ( AC ) s ( BC ) s /ABC ). Moreover, observe that since ϕ ′ ( D ) = acl( AC ), and by definition of Θ , we get that Θ defines a homeomorphism between G al ( L /F ) and G ( D ). Similarly, Θ defines a homeomorphism between G al ( L /F ) and G ( D ).As G al (( AB ) s L L /F ) projects onto G (acl( AB )), onto G al ( L /F ) and onto G al ( L /F ) (viathe restriction maps), we get that U = Θ( G al (( AB ) s L L /F )) projects onto G (acl( AB )), onto G ( D ) and onto G ( D ). Let D be the subfield of ( AB ) s ( AC ) s ( BC ) s fixed by U . Then D isa regular extension of acl( AB ), and of D and D . By Theorem 1.2, there is an elementaryextension F ∗ of F , such that F ∗ ∩ ( AB ) s ( AC ) s ( BC ) s = D . Note that D = acl( AC ) and D = acl( BC ).To finish the proof, we need to show that C realises tp ( C /A ) ∪ tp ( C /B ), and that S G ( C ) = S . Consider ϕ ′ : ( AC ) s → ( AC ) s . Then ϕ ′ ( C ) = C and ϕ ′ (acl( AC )) = acl( AC ) (because F ∗ is a regular extension of D ). We therefore only need to show that the double-dual S Φ ′ is elementary in the structure S G ( F ∗ ). By definition, S Φ ′ = S Ψ ′ − S Θ . By definition of D ,the Galois extensions L F ∗ and D s F ∗ are equal, and S Θ is the identity on G al ( L F ∗ /F ∗ ) = G al ( D s F ∗ /F ∗ ). Also, S Ψ ′ is an elementary isomorphism of S G ( F ), hence also of S G ( F ∗ ). Thisshows that S Φ ′ is elementary, and therefore that tp ( C/A ) = tp ( C /A ). Similarly one showsthat tp ( C/B ) = tp ( C /B ). From the definition of F one also deduces that F ∗ L = F ∗ C s , whichfinishes the proof. Remark/Corollary 2.2. If F is Frobenius, then the condition on the S Ψ i can be relaxed totheir being L G -isomorphisms. The existence of S is still required, for trivial reasons: one couldhave S G ( C ) ⊂ S G ( A ) and S G ( C ) S G ( B ) . Definition 2.3.
Let F be a PAC field, let a , b , c be subsets of F , C = acl( c ), A = acl( c, a )and B = acl( c, b ). We say that a and b are weakly independent over c if A and B are SCF-independent over C and tp ( S G ( A ) /S G ( B )) does not fork over S G ( C ). Remark . Note that this notion is in general not symmetric.
Theorem 2.4.
Let F be a PAC field, let E = acl( E ) ⊂ F , let a , b , c , c be tuples of elementsof F , and let A = acl( Ea ) , B = acl( Eb ) and C i = acl( Ec i ) for i = 1 , . Assume that E containsa p -basis of F if the degree of imperfection of F is finite, and moreover that(i) A ∩ B = E .(ii) A and C are weakly independent over E , B and C are weakly independent over E .(iii) tp ( c /E ) = tp ( c /E ) .(iv) by (iii), there is an E s -isomorphism ϕ : C s → C s , sending c to c and C to C ,such that the double dual map S Φ : S G ( C ) → S G ( C ) is elementary (in S G ( F )) ). As-sume that there is S realising tp ( S G ( C ) /S G ( A )) ∪ tp ( S G ( C ) /S G ( B )) , and such that tp ( S /S G ( A ) , S G ( B )) does not fork over S G ( E ) . (The variables of the two types areidentified via the double dual S Φ of ϕ .) hen there is c realising tp ( c /A ) ∪ tp ( c /B ) , weakly independent from ( a, b ) over E .Proof . Clear by Theorem 2.1. Theorem 2.5.
Let F be a PAC field, and assume that Th( S G ( F )) is simple. Then Th( F ) satisfies the weak independence theorem over submodels, i.e., in the notation of 2.4, if E ≺ F , a and b are weakly independent over E , and a, b, c , c satisfy the hypotheses (ii) – (iii) of 2.4,then there is c realising tp ( c /Ea ) ∪ tp ( c /Eb ) , weakly independent from ( a, b ) over E .Proof . Apply Theorem 2.4: (i) follows from the weak independence of a and b over E , and (iv)because S G ( F ) satisfies the independence theorem over models, by a result of Kim-Pillay [15]. Theorem 2.6.
Let F be a Frobenius field, E ⊂ F . Let a, b, c , c be tuples in F , and A =acl( Ea ) , B = acl( Eb ) , C = acl( Ec ) and C = acl( EC ) . Assume that(i) A ∩ B = E .(ii) a and c are SCF-independent over E , and b and c are SCF-independent over E .(iii) tp ( c /E ) = tp ( c /E ) .(iv) acl( S G ( A )) ∩ acl( S G ( C )) = S G ( E ) , acl( S G ( B )) ∩ acl( S G ( C )) = S G ( E ) .Then there is c realising tp ( c /Ea ) ∪ tp ( c /Eb ) and which is weakly independent from ( b, c ) over E .Proof . By Theorem 2.4 in [2], Th( S G ( F )) is ω -stable, hence simple. By Proposition 4.1 in [2],two subsets of S G ( F ) are independent over the intersection of their algebraic closure. ApplyTheorem 2.5. Remark 2.7.
Condition (iv) is a little bit awkward. It is equivalent to S G ( A ) ∩ S G ( C ) = S G ( B ) ∩ S G ( C ) = S G ( E ) in the following two cases • if F is ω -free, or more generally, is c-Frobenius (see (6.6) in [4] for a definition), • or if E ≺ F . Proof . The first assertion follows from the fact that if F is a c-Frobenius field, then anysubsystem of S G ( F ) is algebraically closed. (In that particular case, the result already appearsin Theorem 6.4 of [4].)To show the second assertion, observe first that S = S G ( E ) is algebraically closed: this isbecause E ≺ F . Proposition 4.1 of [2] tells us that if S, S , S are subsystems of S G ( F ) with S ∩ S = S and S algebraically closed, then S | ⌣ S S , and therefore acl( S ) ∩ acl( S ) = acl( S ) = S . Definition 2.8.
Let n be an integer >
2. A theory T has NSOP n if for every formula ϕ ( x, y )(with x , y of the same length), if M is a model of T , and a i , i ∈ ω , is an infinite sequence ofelements of M such that M | = ϕ ( a i , a j ) whenever i < j , then there are b , . . . , b n in M such that M | = ϕ ( b i , b i +1 ) for i = 1 , . . . , n −
1, and M | = ϕ ( b n , b ). An easy application of compactnessgives the following equivalent formulation: for all m and E ⊂ M , if p ( x, y ) is a 2 m -type oversuch that V i ∈ N p ( x i , x i +1 ) is consistent, then V n − ≤ i p ( x i , x i +1 ) ∧ p ( x n , x ) is also consistent.12 heorem 2.9. Let F be a PAC field, and assume that Th( S G ( F )) has NSOP n for some n > .Then Th( F ) satisfies NSOP n .Proof . If char( F ) = p > F : F p ] < ∞ , we will assume that the language contains constantsymbols for elements of a p -basis. Assume that we have an infinite sequence a i , i ∈ ω , and aformula ϕ ( x, y ) such that ϕ ( a i , a j ) holds whenever i < j . We may assume that the sequence a i is indiscernible, and of length ℵ . By stability of the theory of separably closed fields of agiven degree of imperfection, there is some α < ℵ such that for β ≥ α , tp SCF ( a β /a γ , γ < β )does not fork over E = acl( a γ | γ < α ). Then the sequence a β , β ≥ α , is an infinite sequenceof indiscernibles over E .So, we have reduced to the case where: we have an infinite sequence a i , i ∈ ω , of tuples whichare indiscernible and SCF-independent over E = acl( E ), and which satisfy ϕ ( a i , a j ) whenever i < j ; moreover E contains a p -basis of F if char( F ) = p > F : F p ] < ∞ .For i < j ∈ ω we let A i = acl( E, a i ) and K i,j = acl( A i , A j ). Then for i < j , all tuples( S G ( K i,j ) , S G ( A i ) , S G ( A j )) realize the same L G ( S G ( E ))-type as ( S G ( K , ) , S G ( A ) , S G ( A )).We fix an E s -isomorphism ϕ : A s → A s , which sends A onto A , and denote by S Φ : S G ( A ) → S G ( A ) the double dual.Since Th( S G ( F )) satisfies NSOP n , there is a sufficiently saturated extension F ∗ of F , and S , . . . , S n − ⊂ S G ( F ∗ ) such that ( S i , S i +1 ) and ( S n − , S ) realise tp L G ( S G ( A ) , S G ( A ) /S G ( E ))for 0 ≤ i < n −
1. There are also S i,i +1 , 0 ≤ i < n − S n − , such that the tuples ( S i,j , S i , S j )realise tp L G ( S G ( K , ) , S G ( A ) , S G ( A ) /S G ( E )) for( i, j ) ∈ I := { (0 , , (1 , , . . . , ( n − , n − , ( n − , } . We fix L G ( S G ( E ))-elementary isomorphisms S Ψ i,j : S G ( K , ) → S i,j such that S Ψ i,j ( S G ( A )) = S i , S Ψ i,j ( S G ( A )) = S j , S Ψ i,j | S G ( A ) = S Ψ i,j | S G ( A ) S Φ for i < n , ( i, j ) ∈ I .The strategy is as follows: we will use Lemma 1.15 (and the remark following it) repeatedly tobuild a sequence B , . . . , B n of SCF-independent realisations of tp ( A /E ), with S G (acl( B i B i +1 )) = ( S i,i +1 if i < n − ,S n − , if i = n − , and each pair ( B i , B i +1 ) realises tp ( A , A /E ). We will then apply the amalgamation theorem2.1 to tp ( B /B ) ∪ tp ( B n /B . . . B n − ) to conclude. Let us start the construction:By Lemma 1.15 we find some B realising tp ( A /E ), an E s -isomorphism ψ : A s → B s suchthat ψ ( A ) = B , and the double dual of ψ coincides with the restriction of S Ψ , to S G ( A ).Again, using Lemma 1.15 applied to the extension K , of A , the isomorphisms ψ and S Ψ , ,we now find some B in F ∗ realising ψ ( tp ( A /A )) and SCF-independent from B over E , an E s -isomorphism ψ , : ( A A ) s → ( B B ) s extending ψ , with ψ , ( K , ) = ( B B ) s ∩ F ∗ =: L , ,whose double-dual S G ( K , ) → S G ( L , ) coincides with S Ψ , (so that in particular S G ( L , ) = S , ).Induction step: At stage i < n , we have found B , . . . , B i ⊂ F ∗ which are SCF-independent13ver E , and for each 0 ≤ j < i , E s -isomorphisms ψ j,j +1 : K s , → ( B j B j +1 ) s , with ψ j,j +1 ϕ and ψ j +1 ,j +2 agreeing on A s whenever j < i −
1, such that ψ j,j +1 ( K , ) = ( B j B j +1 ) s ∩ F ∗ , and thedouble dual of ψ j,j +1 coincides with S Ψ j,j +1 on S G ( K , ).We now again apply Lemma 1.15 to the extension K , /A , to the isomorphisms ψ i − ,i ϕ : A s → B si and S Φ i,i +1 : S G ( K , ) → S i,i +1 , and find some B i +1 which is SCF-independent from B · · · B i over E , an isomorphism ψ i,i +1 :( A A ) s → ( B i B i +1 ) s which extends ψ i − ,i ϕ , sends K , to ( B i B i +1 ) s ∩ F ∗ , and such that thedouble dual of ψ i,i +1 coincides with S Ψ i,i +1 .Observe that via S Ψ n − ,n , S G ( B ) = S G ( B n ). By Theorem 2.1, there is B ′ realising tp ( B n /B , . . . , B n − ) ∪ tp ( B /B ) . Then ( B ′ , B , . . . , B n − ) is our desired tuple. Theorem 2.10.
Let F be a Frobenius field. Then Th( F ) satisfies NSOP .Proof . We know that Th( S G ( F )) is ω -stable by Theorem 2.4 in [2], and therefore satisfiesNSOP . The result follows from Theorem 2.9. Theorem 2.11.
Let F be a PAC field, let E , A , B be algebraically closed subsets of F , with E containing a p -basis of F if the degree of imperfection of F is finite, and assume that A and B are weakly independent over E . Then, if B i , i ∈ I , is an indiscernible sequence of realisationsof tp ( B/E ) , which is SCF-independent over E , and if p i denotes the image of tp ( A/B ) by an E -automorphism of F sending B to B i , the type S i ∈ I p i is consistent, and has a realisationwhich is weakly independent from S i ∈ I B i over E .Proof . We may assume that I = N . Using induction and 2.4, one shows that for every n , thereis A ′ realising S i ≤ n p i , weakly independent from S i ≤ n B i . Remark 2.12.
The fact that the B i ’s form an indiscernible sequence over E is completelyunnecessary. We included this hypothesis so as to make it look more like the usual criterionfor forking. What we really prove, is that if the B i ’s are SCF-independent over E , and for all i ∈ I , p i is an extension of tp ( A/E ), having a realisation which is weakly independent from B i over E , then S i ∈ I p i has a realisation which is weakly independent from S i ∈ I B i over E . Comments 2.13.
Theorem 2.10 also follows from results of Chernikov and Ramsey (Theorem6.2 in [7]). Ramsey ([18]) shows that if F is a PAC field with S G ( F ) NTP or NSOP , then F is also NTP or NSOP . Questions 2.14.
We conclude this section with several questions. Throughout, F is a PACfield. I believe that the answer to most questions is positive.(1) (Strengthening of Theorem 2.9) If Th( S G ( F )) does not have the strict order property,then neither does Th( F ).(2) Assume that Th( S G ( F )) satisfies n -amalgamation. Then so does Th( F ) for the weakindependence relation. 143) If F is ω -free, then Th( F ) satisfies n -amalgamation for the strong independence relation.(Recall that A and B are strongly independent over E if they are SCF-independent over E and acl( AB ) = acl( A )acl( B ).)(4) Characterize þ -forking, and whether it is equivalent to þ -forking at the level of the Galoisgroups.(5) Is the theory of ω -free PAC fields rosy? Superrosy?(6) . . . . Let G be a profinite group, SG its associated system. It is convenient toconsider the equivalence relation ∼ associated to ≤ : if α, β ∈ SG we define α ∼ β if and onlyif α ≤ β and β ≤ α . We denote by [ α ] the ∼ -equivalence class of α : it comes with a group lawwith graph given by [ α ] ∩ P , and for β ≥ α , with a group epimorphism, denoted π αβ : [ α ] → [ β ],with graph given by C ∩ ([ α ] × [ β ]). The set of ∼ -equivalence classes, equiped with the inducedpartial ordering ≤ , is a modular lattice with sup ( ∨ ) and inf ( ∧ ). For α, β ∈ SG , we let α ∨ β denote the identity element of [ α ] ∨ [ β ], and α ∧ β the identity element of [ α ] ∧ [ β ]. In otherwords: if N is the kernel of the natural epimorphism G → [ α ] and N the kernel of the naturalepimorphism G → [ β ], then α ∨ β is the identity element of [ α ] ∨ [ β ] = G/N N , and α ∧ β theidentity element of [ α ] ∧ [ β ] = G/N ∩ N .If A is a subsystem of SG and α ∈ SG , we denote by α ∨ A the smallest (for ≤ ) element of { α ∨ β | β ∈ A } . G . If G is a profinite group, then G acts on itself by conjugation. This inducesan action of G on S ( G ), which respects the L G -structure of S ( G ). The action of an element g on a given ∼ -equivalence class G/N is then given by conjugation by gN , and so does notdepend on the choice of the coset representative for gN . This also defines an action of G on allcartesian powers S ( G ) m .Let σ be a tuple of elements of S ( G ), and θ ( ξ, ζ ) an L G -formula. If N is an open subgroupof G such that (the coset) N is ≤ than all the elements of σ , then conjugation by the elementsof N leaves the elements of the tuple σ fixed, so that the set defined by the formula θ ( ξ, σ ) willbe invariant under conjugation by N . Hence, the set defined by θ ( ξ, σ ) will be invariant underconjugation by G if and only if, for all τ ∈ G/N , the sets defined by θ ( ξ, τ − στ ) and by θ ( ξ, σ )coincide. Observe that in any case, the formulas W τ ∈ G/N θ ( ξ, τ − στ ) and V τ ∈ G/N θ ( ξ, τ − στ )(with parameters σ and τ in G/N ) define sets which are invariant under the action of G .One can also also define an action of G n on S ( G ) m × · · · × S ( G ) m n in the natural manner. . Let L be a Galois extension of K of degree n , and let σ , . . . , σ m elements of G al ( L/K ). We say that a tuple of elements of K is a code for ( L, σ , . . . , σ m ) if it is of theform ( a, b , . . . , b m ), where, if a = ( a , . . . , a n ), the polynomial p ( T ) = T n + P ni =1 a n − i T i is the minimal monic polynomial over K of some generator α of L over K , and if b i =15 b i , . . . , b ni ), then σ i ( α ) = P nj =1 b ji α j − . Note that a tuple coding ( L, σ , . . . , σ m ) will alsocode ( L, τ − σ τ, . . . , τ − σ m τ ) for any τ ∈ G al ( L/K ). By abuse of language, we will say that(
L, σ, α, p ( T )) is the data associated to the code ( a, b , . . . , b m ) ( σ = ( σ , . . . , σ m )). We will alsosay that ( a, b , . . . , b m ) codes ( L, σ , . . . , σ m , α ).Note also that the above remark shows that any orbit in G al ( L/K ) m (under the action of G al ( L/K ) is in fact an imaginary of K . S G ( K ) . Let K be a field. While we know ([6]) that the elementaryequivalence of two fields implies the elementary equivalence of the complete systems associatedto their absolute Galois groups, the proof of this result does not show that subsets of S G ( K )are “definable over K ”. This is easy to see: let S ⊂ S G ( K ) m be definable over G al ( L/K ), andlet τ ∈ G ( K ): then τ leaves K fixed, but its double dual sends S to τ Sτ − . One however hasthe following result: Proposition 3.5. (Cherlin-van den Dries-Macintyre [6]). Given an L G -formula θ ( ξ ) , whichin particular says that the elements of the tuple ξ live in the same ∼ -equivalence class, there isa formula of the language of fields θ ∗ ( x ) such that for any field K , and code a for an ( L, σ ) ofthe appropriate sort, K | = θ ∗ ( a ) ⇐⇒ S G ( K ) | = θ ( σ ) . The proof is easy: if tp ( b ) = tp ( a ), there is an automorphism of some ultrapower K U of K which sends a to b . This automorphism extends to an automorphism of ( K U ) s , with doubledual sending σ to a tuple σ ′ coded by b . Then tp ( σ ) = tp ( σ ′ ) (in S G ( K U ) and therefore in S G ( K )). Hence, if σ satisfies θ ( ξ ), there is some formula θ a ( x ) ∈ tp ( a ) which “implies” θ , i.e.,such that if b satisfies θ a , then any tuple σ ′ coded by b will satisfy θ . By compactness we getthe formula θ ∗ ( x ).The difficulties occur when one deals with an arbitrary L G -formula θ ( ξ ), and in general onecannot hope for a similar result. The problem is that if σ = ( σ , . . . , σ n ) and a i is a codefor ( L i , σ i ), then a i only defines σ i up to conjugation by the elements of G al ( L i /K ). Thusalready a formula of the form ξ i = ξ j poses problem: one cannot expect to have a formula θ ( x i , x j ) which expresses this property of all elements coded by x i and x j . This problemcan be addressed by adapting the definition of codes, however is quite unpleasant to formulatein the general case. Here, we will deal with a particular case. Definition 3.6. (1) Let L be a finite Galois extension of K , σ a tuple of elements in G al ( L/K ),and α, β ∈ L such that L = K ( α ). We say that a codes ( L, σ, α, β ) if it is of the form( b, c ) where b is a code for ( L, σ, α ) (see 3.3), and c gives the coordinates of β with respectto the basis { , α, . . . , α [ L : K ] − } of L over K .(2) Let L and L be finite Galois extensions of a field K , σ and σ tuples of elementsin G al ( L /K ), G al ( L /K ) respectively, and L = L ∩ L . Let α , α , α be such that L i = K ( α i ), i = 0 , ,
2. We say that ( a , a ) is a 2 -code for ( L , L , σ , σ , α , α , α ) if a is a code for ( L , σ , α , α ) and a is a code for ( L , σ , α , α ).(3) We say that a is a -code for ( L , L , σ , σ ) if it is a 2-code for ( L , L , σ , σ , α , α , α )for some α , α , α . 164) An L G -formula θ ( ξ ) is codable if it implies that the elements of the tuple ξ are ∼ -equivalent.(5) An L G -formula θ ( ξ , ξ ) is 2 -codable if it implies that the elements of the tuple ξ i are ∼ -equivalent, for i = 1 , Remark 3.7.
One checks easily that being a 2-code of some ( L , L , σ , σ ) (with [ L L : K ] ≤ n for a fixed n ), is an elementary property of a tuple (again, one uses that a tuple havingthe same type as a 2-code, is a 2-code; see also 3.9). One also notes that if ( a , a ) is a 2-code for ( L , L , σ , σ ), then ( a , a ) codes exactly the tuples ( L , L , ρ | L σ ρ | L − , ρ | L σ ρ | L − )where ρ ∈ G al ( L L /K ). Indeed, assume that ( a , a ) codes ( L , L , σ , σ , α , α , α ) and( L , L , τ , τ , β , β , β ). Then β = ρ ( α ) for some ρ ∈ G al ( L /K ), and β = ρ ( α ), τ = ρ σ ρ − . Similarly, β = ρ ( α ) for some ρ ∈ G al ( L /K ), and β = ρ ( α ) , τ = ρ σ ρ − . Thisimplies that ρ and ρ agree on L = L ∩ L , and that they can be extended to a common ρ ∈ G al ( L L /K ). Proposition 3.8.
Let θ ( ξ , ξ ) be a -codable L G -formula. There is a formula θ ∗ ( x , x ) of thelanguage of fields, such that in any field K , if ( a , a ) is a -code for ( L , L , σ , σ ) , then K | = θ ∗ ( a , a ) ⇐⇒ S G ( K ) | = θ ( σ , σ ) . Proof . Reason as in the proof of Proposition 3.4. . We saw earlier that being a code, or a 2-code, is anelementary property. If one wishes to show this result more explicitly, one needs to work alittle.To express that ( a , a ) is a code for ( L, σ ) is fairly easy. One says first of all that the monicpolynomial p ( T ) whose coordinates are given by a is irreducible over K and separable. Then,if α is a root of p ( T ), one can interpret in K the pair of fields ( K ( α ) , K ), by identifying K ( α )with K ⊕ Kα ⊕ · · · ⊕ Kα n − where n is the degree of p ( T ). One then says that K ( α ) containsall n roots of p ( T ), and that the tuple a consists of coordinates of some of these roots (indeed,one can code an element τ of G al ( K ( α ) /K ) by specifying the coordinates of τ ( α )).We now want to express the fact that (( a , a , a ) , ( b , b , b )) is a 2-code. That ( a , a , a )is a code for some ( L, σ, α, γ ) is expressible, follows from the previous paragraph, and similarlythat ( b , b , b ) is a code for some ( M, τ, β, δ ). As before, one can interpret in K , using the pa-rameters ( a , a , a ) the structure ( L, K, σ, α, γ ), and similarly, using the parameters ( b , b , b ),the structure ( M, K, τ, β, δ ). To express that (( a , a , a ) , ( b , b , b )) is a 2-code, we need toexpress the following:– that γ and δ are conjugates over K .– that L ∩ M = K ( γ ) = K ( δ ).The first item is easy: in the structure ( K ( α ) , K, α, γ ), one can define the coefficients ofthe minimal (monic) polynomial r ( T ) of γ over K , and similarly one can define in ( M, K, β, δ )the coefficients of the minimal polynomial of δ over K . It then suffices to say that these twominimal polynomials are the same, and that K ( γ ) contains all roots of r ( T ).For the second item, observe that in fact the triples ( L, K ( γ ) , K, α, γ ) and ( M, K ( δ ) , K, β, δ )are interpretable from ( a , a , a ) and ( b , b , b ). In ( L, K ( γ ) , K, α, γ ), the coefficients of theminimal polynomial q ( γ, T ) of α over K ( γ ) are definable. It therefore suffices to say that q ( δ, T )17s irreducible over M , and this is expressible in the structure ( M, K, β, δ ). [The first item givesus that L ∩ M ⊇ K ( γ ). The irreducibility of q ( T, δ ) over M implies that [ L : K ( δ )] = [ LM : M ],so that L ∩ M = K ( δ )].All this is done uniformly in the length of the parameters involved, and so gives the first-order expressibility of “( x, y ) is a 2-code”. Note however that the partition of the variables ofthe formula needs to be fixed, i.e., one needs to know [ L : K ] = n , | σ | = i , [ M : K ] = m and | τ | = j : if x = ( x , . . . ) then a will correspond to ( x , . . . , x n ), a to ( x n +1 , . . . , x n ( i +1) ), and a to ( x n ( i +1)+1 , . . . , x n ( i +2) ), and similarly for the elements of the tuple y . . Let ρ , . . . , ρ m enumerate an ∼ -equivalence class of S ( G ). Thenthe elements of the subsystem of S ( G ) generated by ρ are in the definable closure of ρ , . . . , ρ m .Indeed, each τ ∈ h ρ i is ≥ ρ ; consider the set I ( τ ) of indices j such that C ( ρ j , τ ) holds. Becausethe ρ i enumerate the ∼ -class of ρ , the element τ is uniquely defined by the formula ^ j ∈ I ( τ ) C ( ρ j , ξ ) ∧ ^ j / ∈ I ( τ ) ¬ C ( ρ j , ξ ) . Notation 3.11.
Let S , S be subsets of S ( G ). We denote by tp ( S /S ) the set of all formulasof the form θ ( ξ , ξ ) ∈ tp ( S /S ), where the L G -formula θ ( ξ , ξ ) is 2-codable.Let K be a field, and A , B subfields of K such that F is a regular extension of A and of B .We denote by tp ∗ ( S G ( A ) /S G ( B )) the set of formulas θ ∗ ( X, B ), where θ (Ξ , Ξ ) is a 2-codableformula, and θ (Ξ , S G ( B )) ∈ tp ( S G ( A ) /S G ( B )). Lemma 3.12.
Let S and S be subsystems of S ( G ) , and assume that S satisfies tp ( S /S ) .Then tp ( S /S ) = tp ( S /S ) .Proof . By compactness, we may assume that S , S and S are finite. For i = 1 ,
2, let σ i be anenumeration of the smallest (for ≤ ) ∼ -equivalence class of S i , and let σ be the subtuple of S corresponding to σ ⊂ S . By assumption, tp ( σ /σ ) = tp ( σ /σ ), and by Observation 3.10 weget the result. Remarks 3.13. (1) A finite disjunction of 2-codable formulas is not necessarily 2-codable,but a result analogous to 3.8 holds nevertheless:Let θ i ( ξ i , ζ i ) be 2-codable formulas, i = 1 , . . . , n . Then for every field K , and codes ( a i , b i )for ( L i , M i , σ i , τ i ), we have K | = n _ i =1 θ ∗ i ( a i , b i ) ⇐⇒ S G ( K ) | = n _ i =1 θ i ( σ i , τ i ) . (2) This result becomes false if one replaces the disjunctions by conjunctions. However, notethat a Boolean combination of 2-codable formulas in the same variables is 2-codable.(3) From Lemma 3.12, it follows that if the field K is κ -saturated, then so is S G ( K ). Lemma 3.14.
Let S and S be subsystems of some S ( G ) , and let Ξ , Ξ enumerate the vari-ables of qf tp ( S ) and qf tp ( S ) (the quantifier-free types). Let θ ( ξ , ξ ) be a Boolean combinationof -codable formulas, ξ i ⊂ Ξ i . Then there is a -codable formula θ ′ ( ζ , ζ ) , ( ζ i ⊂ Ξ i ) such that qf tp ( h ζ i ) ∪ qf tp ( h ζ i ) ⊢ θ ( ξ , ξ ) ↔ θ ′ ( ζ , ζ ) . roof . Say that θ ( ξ , ξ ) is a Boolean combination of the 2-codable formulas θ i ( ξ i , ξ i ). Let ζ ⊂ Ξ enumerate a ∼ -equivalence class such that qf tp ( S ) ⊢ V i ( ζ ≤ ξ i ), and let ζ bedefined similarly for ξ i . By Observation 3.10, there are 2-codable formulas θ ′ i ( ζ , ζ ) such that qf tp ( S ) ∪ qf tp ( S ) ⊢ θ i ( ξ i , ξ i ) ↔ θ ′ i ( ζ , ζ ) . Any Boolean combination of the θ ′ i ( ζ , ζ ) is 2-codable, and this gives the result (get rid of theextra variables of qf tp ( S i ).). Definition 3.15.
Let θ ( ξ ) be a codable formula of L G . We define θ ∗ ( x ) to be the formula ofthe language of fields which satisfies the following condition, in any field K : For any tuple a in K , K | = θ ∗ ( a ) if and only if a is a code for some ( L, σ ) , and S G ( K ) | = θ ( σ ) . Similarly, if θ ( ξ , ξ ) is a 2-codable formula, we let θ ∗ ( x, y ) be the formula of the language offields which satisfies the following, for every field K : For any tuple ( a, b ) in K , K | = θ ∗ ( a, b ) if and only if ( a, b ) is a -code for some ( L, M, σ, τ ) and S G ( K ) | = θ ( σ, τ ) . The formulas θ ∗ ( x ) and θ ∗ ( x, y ) exist, by the discussion above and by 3.4, 3.8. Note that( ¬ θ ) ∗ = ¬ ( θ ∗ ). tp ∗ . Let K be a field, and A , B subfields of K such that K ∩ A s = A , K ∩ B s = B . We denote by tp ∗ ( S G ( B ) /S G ( A )) the set of formulas θ ∗ ( X , A ), where θ (Ξ , Ξ ) ∈ tp ( S G ( B ) /S G ( A )), and θ ∗ ( X , X ) is the formula of the language of fields associ-ated to θ (Ξ , Ξ ) as in the above definition 3.15.Here we need a word of explanation about the variables. The elements of X correspond to anenumeration of B , and similarly for the variables of X . That C satisfies tp ∗ ( S G ( B ) /S G ( A ))will mean that we have fixed an enumeration of C corresponding to the elements of X . Fromthe definition of the formulas θ ∗ ( X , X ) we obtain the following: Remarks 3.17.
Assume that C satisfies tp ∗ ( S G ( B ) /S G ( A )).(1) There is an elementary S G ( A )-isomorphism f : S G ( C ) → S G ( B ), which respects thecoding, i.e., if a ⊂ A , b ⊂ B are such that ( b, a ) is a 2-code for some ( L , L , σ, τ ),and if c ⊂ C is the subtuple of C corresponding to b ⊂ B , then ( c, a ) is a 2-code for( L , L , f ( σ ) , τ ), where L is defined by G al ( L /B ) = f ( G al ( L /C )).(2) If the correspondence between the elements of B and the elements of C (given by X ) de-fines a field isomorphism, then f is in fact induced by some extension of this isomorphismto an A s -isomorphism with domain B s . In this section, we will show how the type amalgamation result gives information about imag-inaries. In the later part of this chapter, we will fix a large PAC field F , of characteristic p .If p >
0, then we assume that its degree of imperfection is finite, and add to the languageof rings constant symbols for elements of a p -basis. All our subfields of F will contain thesedistinguished elements. This has two consequences which we will use:191) The theory of separably closed fields in this expanded language, together with axiomssaying that the new constants form a p -basis, is complete and eliminates imaginaries([8]).(2) If A and B are subfields of F closed under the λ -functions of F (which give the coordinatesof elements with respect to the fixed p -basis), then AB is also closed under the λ -functionsof F . This implies (4.5 in [5]) that acl( AB ) = F ∩ ( AB ) s .Before starting with the description of imaginaries, we will take a closer look at subsets of S G ( F ) which are definable in F . Definition 4.1. A basic imaginary of F is a pair ( a, D ), where a is a tuple of elements of F ,and D is a definable subset D of S G ( F ) m for some m , which is stable by conjugation under theelements of G ( F ).Here, for us, a definable or interpretable set is an imaginary element, i.e., we identifydefinable/interpretable sets with their codes in the sense of S G ( F ) eq . Thus, if L is a finiteGalois extension of F , then G al ( L/F ) is an imaginary, and so is L .It is clear from the discussion in section 3 that basic imaginaries are indeed imaginariesof the field F . The requirement that the set D be stable under conjugation is necessary, aselements of G ( F ) will fix elements of F eq . Theorem 4.2.
Let F be a PAC field, of finite degree of imperfection if the characteristic ispositive, and expand the language by adding constants for elements of a p -basis if necessary.Let e ∈ F eq . Then e is equi-definable with a finite set of basic imaginaries.Proof . The proof is fairly long, and proceeds with a series of steps. We will assume that F issufficiently saturated. Let E = acl eq ( e ) ∩ F , E = dcl eq ( e ) ∩ F . Then E is a Galois extensionof E , and every element of G al ( E/E ) lifts to an automorphism of F eq fixing e .If e ∈ acl eq ( E ), we are done: e is coded by a tuple of elements of E . We will therefore assumethat this is not the case, and fix a 0-definable map f , and a tuple a such that f ( a ) = e . We let A = acl( E, a ), and consider the set P of realisations of tp ( A/E ). We also write f ( A ) = e . Thefirst step is by now a routine argument.We will consider the fundamental order on types (in the sense of Th( F s )), denoted by ≤ fo ,and ∼ fo will denote the associated equivalence relation. We refer to chapter 13 of [17] for thedefinition and properties of the fundamental order. Recall that if p and q are stationary types,then p ∼ fo q iff they have a common non-forking extension. In our setting, we have that if D = acl( D ) and d is a tuple in F , then tp SCF ( d/D ) is stationary. And of course, any type inthe sense of Th( F s ) over a separably closed field is stationary. Step 1 . There is B ∈ P , with f ( B ) = e , and which is SCF-independent from EA over E .By Lemma 1.4 of [11], there is B realising tp ( A/ acl eq ( E, e )) such that acl eq ( E, B ) ∩ acl eq ( E, A ) =acl eq ( E, e ), whence f ( B ) = e and acl( E, A ) ∩ acl( E, B ) ∩ F = E. ( ∗ )Observe that because A is a regular extension of E , tp SCF ( A/E ) is stationary, and so is everynon-forking extension. Consider the set B of realisations of tp ( A/ acl eq ( E, e )) which satisfy ( ∗ ),20nd choose B ∈ B such that tp SCF ( B/ ( EA ) s ) is maximal for the fundamental order in the set { tp SCF ( B ′ / ( EA ) s ) | B ′ ∈ B} . Let C realise tp ( B/ acl( EA )), SCF-independent from B over EA .(Note that e ∈ acl( EA ), and so C realises tp ( A/ acl eq ( E, e )).) Then tp SCF ( C/ ( EAB ) s ) ∼ fo tp SCF ( B/ ( EA ) s ) , and f ( C ) = e. Since tp SCF ( C/ ( EAB ) s ) ≤ fo tp SCF ( C/ ( EB ) s ), we obtain tp SCF ( B/ ( EA ) s ) ≤ fo tp SCF ( C/ ( EB ) s ) . Moreover,acl( EB ) ∩ acl( EA ) ∩ F ⊆ acl( EAB ) ∩ acl( EC ) ∩ F = acl( EA ) ∩ acl( EC ) ∩ F = E, so that the pair ( B, C ) satisfies ( ∗ ). Since B and C both realise tp ( A/ acl eq ( E, e )), the maxi-mality of tp SCF ( B/ ( EA ) s ) among the extensions of tp SCF ( A/ acl eq ( E, e )) satisfying ( ∗ ) and theinequality tp SCF ( B/ ( EA ) s ) ≤ fo tp SCF ( C/ ( EB ) s )imply that tp SCF ( B/ ( EA ) s ) ∼ fo tp SCF ( C/ ( EB ) s ) . Hence tp SCF ( C/ ( EAB ) s ) ∼ fo tp SCF ( B/ ( EA ) s ) ∼ fo tp SCF ( C/ ( EB ) s ) , and tp SCF ( C/ ( EAB ) s ) does not fork over EB . By elimination of imaginaries in SCF (re-call that the degree of imperfection is finite, and that E contains a p -basis), we obtain that tp SCF ( C/EAB ) does not fork over acl
SCF ( EA ) ∩ acl SCF ( EB ) = E s , and therefore does notfork over E . Since tp ( C/ ( EA ) s ) = tp ( B/ ( EA ) s ), we have tp SCF ( B/EA ) does not fork over E ,which proves the result. Step 2 . Let B ∈ P , SCF-independent from a over E , and with f ( B ) = e . Assume that C ∈ P is SCF-independent from B over E , and that there is an E s -isomorphism ϕ : B s → C s , whosedouble dual S Φ : S G ( B ) → S G ( C ) is an L G ( S G ( A ))-elementary map. Then f ( C ) = e .We will apply the results of Theorem 2.1, to show that tp ( B/A ) ∪ tp ( C/B ) is consistent (theidentification between the variables of tp ( B/A ) and of tp ( C/B ) being given by the fact thatthese types extend tp ( A/E )). In the notation of this theorem, we let ϕ = ϕ , S = S G ( C ), C = A , C = C , S Ψ is the identity of h S G ( C ) , S G ( B ) i , and S Ψ is the partial (elementary)isomorphism on h S G ( B ) , S G ( A ) i extending S Φ and the identity of S G ( A ).If C ′ realises tp ( B/A ) ∪ tp ( C/B ), then f ( C ′ ) = f ( B ) = e , and this implies that f ( C ) = e . Step 3 . Let
B, C ∈ P with f ( B ) = e . Assume that there is an E s -isomorphism ϕ : B s → C s ,whose double dual S Φ : S G ( B ) → S G ( C ) is an L G ( S G ( A ))-elementary map. Then f ( C ) = e .By Step 1, there is B ′ ∈ P , with f ( B ′ ) = e , and which is SCF-independent from A over E . As tp ( B ′ /A ) has a realisation which is SCF-independent from C over A , we may assume that B ′ is SCF-independent from AC over E . Apply Step 2 to A, B ′ , C . Step 4 . There is a 2-codable formula θ (Ξ , Υ) such that, if C ∈ P , then f ( C ) = e if and onlyif S G ( C ) satisfies θ (Ξ , S G ( A )), if and only if C satisfies θ ∗ ( X, A ).21ere we need a word about the variables. If B ∈ P , then by definition we have an E -isomorphism ϕ : B → A (which is elementary). This isomorphism extends to an E s -isomorphism B s → A s , whose double dual is an S G ( E )-isomorphism S G ( B ) → S G ( A ).For each B ∈ P such that f ( B ) = e , we know by Step 3 and by Lemma 3.12 that tp ( B/E ) ∪ tp ( S G ( B ) /S G ( A )) ⊢ f ( X ) = e. Hence there is θ B (Ξ , Υ) such that θ B (Ξ , S G ( A )) ∈ tp ( S G ( B ) /S G ( A )) and tp ( B/E ) ∪ θ ∗ B ( X, A ) ⊢ f ( X ) = e. By compactness, a finite disjunction of the θ ∗ B ( X, A ) is equivalent to f ( X ) = e modulo tp ( B/E ).By Lemma 3.14, we may replace this disjunction by θ ∗ ( X, A ) for some 2-codable formula θ (Ξ , Υ).
Step 5 . Let
B, C ∈ P . Then C satisfies θ ∗ ( X, B ) if and only if f ( C ) = f ( B ).Indeed, there is an E -automorphism of F which sends A to B . Then the elements of P satisfying θ ∗ ( X, B ) are precisely those satisfying f ( X ) = f ( B ). Step 6 . There is a set D , definable over S G ( A ), such that an E -automorphism σ of F fixes e if and only some (any) extension ˜ σ of σ to F s leaves D invariant.We know by Step 4 that tp ( A/E ) ⊢ θ ∗ ( X, A ) ⇐⇒ f ( X ) = e. Let σ ∈ Aut(
F/E ) fix e , and ˜ σ an extension of σ to F s . Then σ induces an automorphism of theset P , which leaves invariant the set of realisations of θ ∗ ( X, A ), by step 4. Hence, (conjugationby) ˜ σ leaves invariant the set D of realisations of θ (Ξ , S G ( A )). I.e., D ∈ dcl eq ( E, e ).Conversely, let σ ∈ Aut(
F/E ), and ˜ σ an extension of σ to F s which leaves D invariant. Then˜ σ leaves invariant the set P , as well as any L S G ( E ) -definable subset of D which is stable byconjugation. Hence it leaves invariant the set of realisations of θ (Ξ , S G ( A )) which are a subtupleof some realisation of tp ( S G ( A ) /S G ( E )). By Lemma 1.15, there is some B ∈ P such that S G ( B ) satisfies θ (Ξ , S G ( A )). I.e., B satisfies θ ∗ ( X, A ) and f ( B ) = e . So e ∈ dcl eq ( E, D ). Step 7 . The result.It follows that the imaginary D and e are equi-definable over E . Hence there is a finite tuple a of elements of E such that they are equi-definable over a . Then e is equi-definable with theset of conjugates of ( a, D ) over e . . This result is not totally satisfactory: it would have been better to obtain asingle basic imaginary. We will show by an example below that it is not always possible. One canhowever observe that e can be squeezed between two basic imaginaries: namely e ∈ dcl eq ( a, D ),and if b codes the set of conjugates of a over e , and D ′ the set of conjugates of D over e , then e is algebraic over ( b, D ′ ).It turns out that the interaction between F and S G ( F ) at the level of algebraic closure is veryweak: Proposition 4.4.
Let F be a PAC field. If F is of characteristic p > , we assume that itsdegree of imperfection is finite and that we have constant symbols for elements of a p -basis of F . Let e = { ( a , D ) , . . . , ( a n , D n ) } ∈ F eq , where each ( a i , D i ) is a basic imaginary. eq ( e ) ∩ F = acl SCF ( a , . . . , a n ) ∩ F (= acl( a , . . . , a n ) in the sense of the theory of F ). (2) acl eq ( e ) ∩ S G ( F ) = acl eq ( S G (acl( a , . . . , a n ) , D , . . . , D n )) ∩ S G ( F ) . Here, as before,by D , . . . , D n , we mean the elements of S G ( F ) eq corresponding to the sets defined by D , . . . , D n .Proof . (1) Using 3.10, we may assume that all the D i ’s are definable over G al ( L/F ), for somefinite Galois extension L of F . Let A = acl( a , . . . , a n ), b any finite tuple of elements of F \ A such that A ( b ) contains a code of the extension L and of the elements of G al ( L/F ).Let B = acl( A, b ), and choose an A s -automorphism ϕ of Ω such that ϕ ( B ) = C is linearlydisjoint from F over A . Let Φ : G ( B ) → G ( C ) be the dual of ϕ − , and consider the subgroup H = { ( σ, Φ( σ | B s )) | σ ∈ G ( F ) } . Let M be the subfield of C s F s fixed by H . Then M is a regularextension of C and of F , and the restriction map G al ( C s F s /M ) → G ( F ) is an isomorphism.By Theorem 1.2, F has an elementary extension F ∗ containing M , regular over M . Then tp F ∗ ( C/A ) = tp F ∗ ( B/A ) = tp F ( B/A ). By definition of H , the tuple of G al ( L/F ) coded by ϕ ( b )is the same (up to conjugation) as the tuple coded by b . Moreover, as b was any finite tuple of F \ A , and ϕ ( b ) = b , it follows that b / ∈ acl eq ( e ).(2) Let S = acl eq ( S G ( A ) , D , . . . , D n )) ∩ S G ( F ). Going to some sufficiently saturated extension F ∗ of F and using again Lemma 1.4 of [11], we find S ′ realising tp ( S G ( F ) /S ) and such that S ′ ∩ S G ( F ) = S . (Note that both S ′ and S G ( F ) are algebraically closed). We fix some L G -elementary map S Ψ : S ′ → S G ( F ) which is the identity on S . By Lemma 1.15, there is F realising tp ( F/A ), and an A s -isomorphism ψ : F s → F s , with double dual S Ψ. Since ψ is theidentity on A s , and S Ψ is the identity on S , it follows that ψ ( e ) = e . This shows (2). . Recall that a Frobenius fieldis a PAC field F , whose absolute Galois group G ( F ) has the embedding property, see see 1.9.The properties of S G ( F ) we will use are the following (see [2], sections 2 and 4): • Th( S G ( F )) is ω -stable. • (Description of the types) Let β , γ be tuples of elements of the equivalence class [ β ], [ γ ]respectively, let S be a subsystem of S G ( F ), and let δ = β ∨ S . Then tp ( β/S ) = tp ( γ/S )if and only if γ ∨ S = δ , and there is an isomorphism f : [ β ] → [ γ ] such that f ( β ) = γ ,and π β,δ = π γ,δ f (i.e., f induces the identity on [ δ ]). • If S is a subsystem of S G ( F ), then the quantifier-free type of S implies its type. • Let A be a subsystem of S G ( F ), and α, β ∈ S G ( F ), with α = β ∨ A . Then β ∈ acl( A ) ifand only if β ∈ acl( α ).One cannot expect the theory of a complete system SG to eliminate imaginaries, simply becausemost finite groups do not eliminate imaginaries: consider for instance Z / Z . Then its subset { , } cannot be coded by any finite tuple of elements of Z / Z . However, in case of profinitegroups with the embedding property, one obtains the next best thing: weak elimination ofimaginaries. 23 heorem 4.6. Let G be a profinite group with the embedding property, SG its associated system.Then Th( SG ) weakly eliminate imaginaries. Furthermore, any imaginary is equi-definable withan imaginary of the form ([ α ] , ε ) , where α ∈ SG , and ε is an imaginary of the group [ α ] .Proof . Let D be a definable subset of SG m , defined over some algebraically closed subsystem A . By Observation 3.10, we may assume that if the m -tuple β is in D , then all its elementsare ∼ -equivalent: indeed, if if ( σ , . . . , σ m ) ∈ D , then there are only finitely many possibleisomorphism types of the system ( h σ , . . . , σ m i , σ , . . . , σ m ); hence there is a definable set D ′ satisfying the required hypothesis, and a finite-to-one onto definable map D ′ → D ; then in SG eq , D ′ ∈ acl( D ). Since Th( SG ) is ω -stable (by Theorem 2.4 in [2]), D contains only finitemany types of maximal Morley rank, say p i = tp ( β i /A ), i = 1 , . . . , r . Then, each p i is definableover [ α i ], where α i = β i ∨ A : if B is a subsystem of SG containing A , then the unique non-forking extension of p i to B is given by p i | [ α i ] ∪ {¬ ( ξ ≥ γ ) | γ ∈ B, γ < α i } . I.e., [ α i ] is the(algebraic closure of a) canonical base for p i . This shows weak elimination of imaginaries. Thelast assertion follows immediately from our first reduction. Theorem 4.7.
Let F be a Frobenius field, of finite invariant if the characteristic is positive,and in that case assume that the language contains symbols of constants for a p -basis of F .Then every imaginary of F is equidefinable with a finite set of basic imaginaries ( a, D ) , wherefurthermore D is an imaginary of some group [ α ] .Proof . This is clear from the discussion above and Theorem 4.2. . Let a, b, c, d be elements which are algebraically independent over Q , and ζ a primitive 3-rd root ot 1. We fix cubic roots α := √ c + a , β := √ c + b , α := √ d + a , β := √ d + b of ( c + a ), ( c + b ), ( d + a ) and ( d + b ) respectively. We then let E = Q ( a, b ) alg , E = E ( c, d, α β , β α )and let F be an ω -free PAC field which is a regular extension of E . Then any automorphismof E , or of E , is elementary in the sense of Th( F ).Let L = F ( α , β ), and σ ∈ G al ( L/F ) be defined by σ ( α ) = ζ α and σ ( β ) = ζ β . Then F ( α ) = F ( β ) and F ( β ) = F ( α ), σ ( α ) = ζ α , σ ( β ) = ζ β . Consider the basic imaginaries e := ( a, ( L, σ )) and e = ( b, ( L, σ )). Note that because G al ( L/F ) is abelian, we do not haveto worry about conjugation. Consider the imaginary e := { e , e } . Then e ∈ dcl eq ( a, b, ( L, σ )),and letting f = ( ab, a + b ) , f = ( L, { σ, σ } ) , we have f , f ∈ dcl eq ( e ). We will show that e is not equidefinable with any basic imaginary( E, ε ). Assume by way of contradiction that dcl eq ( e ) = dcl eq ( E, ε ). Then E = dcl eq ( e ) ∩ F . Claim . E = Q ( f ).We know by Proposition 4.4 that E ⊂ acl( a, b ) = E . Let ρ be any automorphism of E which fixes c, d and ζ , and exchanges a and b . Then ρ ( e ) = e : Indeed, ρ extends toan automorphism ρ ′ of L := E ( α , β ), which sends ( α , β ) to ( β , α ); one then computesthat ρ ′ σρ ′ − = σ . Clearly ρ ( L ) = L , and so, ρ ( e ) = e . This being true for any ρ ∈G al ( E / Q ( f , ζ , c, d )), we get that E ⊆ Q ( f , ζ ). Consider now any automorphism ρ of E a, b , exchanges c and d , and sends ζ to ζ . Then again one computes that ρ ( e ) = e .As ρ moves ζ and fixes f , we obtain that E = Q ( f ).By Proposition 4.4(2) and because ε ∈ dcl eq ( a, b, ( L, σ )),acl( e ) eq ∩ S G ( F ) = acl eq ( S G (acl( a, b )) , ε ) = acl eq ( G al ( L/F )) = hG al ( L/F ) i . (The second equality is because S G (acl( a, b )) = 1, and the third because any subsystem of S G ( F ) is algebraically closed.)However, there is an automorphism ρ of E , which exchanges a and b , exchanges c and d ,and is the identity on ζ . One computes that it induces the identity on G al ( L/F ), and therefore ρ ( e ) = e . Note that this ρ is the identity on E , and so ε / ∈ dcl eq ( E, hG al ( L/F ) i ), which givesus the desired contradiction. References [1] J. Ax, The elementary theory of finite fields, Annals of Math. 88 (1968), 239 – 271.[2] Z. Chatzidakis, Model theory of profinite groups having IP, Illinois J. of Math. 42 No 1(1998), 70 – 96.[3] Z. Chatzidakis, Simplicity and Independence for Pseudo-algebraically closed fields, in:Models and Computability, S.B. Cooper, J.K. Truss Ed., London Math. Soc. Lect. NotesSeries 259, Cambridge University Press, Cambridge 1999, 41 – 61.[4] Z. Chatzidakis, Properties of forking in ω -free pseudo-algebraically closed fields, J. of Symb.Logic 67 Nr 3 (2002), 957 – 996.[5] Z. Chatzidakis, A. Pillay, Generic structures and simple theories, Ann. Pure Applied Logic95 (1998), 71 – 92.[6] G. Cherlin, L. van den Dries, A. Macintyre, Decidability and Undecidability Theorems forPAC-Fields, Bull. of the AMS 4 (1981), 101-104.[7] A. Chernikov, N. Ramsey, On model-theoretic tree properties, J. Math. Log. 16 No. 2(2016), 1650009. DOI: 10.1142/S0219061316500094.[8] F. Delon, Id´eaux et types sur les corps s´eparablement clos, Suppl´ement au Bull. de laS.M.F, M´emoire 33, Tome 116, 1988.[9] J. -L. Duret, Les corps faiblement alg´ebriquement clos non s´eparablement clos ont lapropri´et´e d’ind´ependance, in: Model theory of Algebra and Arithmetic, Pacholski et al.ed., Springer Lecture Notes 834 (1980), 135 –157.[10] Yu. L. Ershov, Undecidability of regularly closed fields (Russian) Algebra i Logika 20(1981), no. 4, 389 – 394, 484.[11] D.M. Evans, E. Hrushovski, On the automorphism group of finite covers, APAL 62 (1993),83 – 112. 2512] M. Fried, M. Jarden, Field Arithmetic , Ergebnisse 11, 3rd edition, Springer Berlin-Heidelberg 2008.[13] E. Hrushovski, Pseudo-finite fields and related structures, in:
Model Theory and Appli-cations , B´elair et al. ed., Quaderni di Matematica Vol. 11, Aracne, Rome 2005, 151 –212.[14] E. Hrushovski, A. Pillay, Groups definable in local fields and pseudo-finite fields, Israel J.of Math. 85 (1994), 203 –262.[15] B. Kim, A. Pillay, Simple theories, APAL 88 Nr 2-3 (1997), 149 – 164.[16] S. Lang,
Introduction to algebraic geometry , Addison-Wesley Pub. Co., Menlo Park 1973.[17] B. Poizat,
A Course in Model Theory , Universitext, Springer-Verlag New York BerlinHeidelberg 2000.[18] Nick Ramsey,
Independence, Amalgamation, and Trees , PhD thesis, Berkeley May 2018.Current address:D´epartement de Math´ematiques et ApplicationsEcole Normale Sup´erieure45 rue d’Ulm75230 Paris Cedex 05Francee-mail: [email protected]@dma.ens.fr