We now review how to translate statements about the inverse system S(Gal(F)) of the absolute Galois group of a field into statements about the field itself. Such a translation is claimed already in the unpublished [6], but without proof. In fact, the reader might be tempted to assume that one can prove [6, Lemma 17] by giving an honest interpretation of theω-sorted structureS(Gal(F))ωinF, but experts think that this is actually impossible.
On the other hand, the abstract proof presented in [4] is complete, but does not allow one to conclude that the translation is recursive. For this section, we fixS= ∅and let Lco=Lco,∅.
Definition C.1. As explained for example in [6,§1], [4, (5.10)], and [1, Proposition 5.1], one can encode finite extensions of F of degree n by n3-tuples a∈Fn3. We denote the extension defined bya∈ Fn3 byFa. Ifa1∈Fn31 anda2∈Fn32 encode extensions of degree n1 (respectively, n2) then an F-embedding of Fa1 into Fa2 is encoded by ann1n2-tuple b∈ Fn1n2.
Definition C.2. Anadmissiblesequence of lengthλωand degrees(ni)i<λis a sequence a=a1a2. . . of elements of F such that, fori < λ, the tupleai encodes
(1) a finite Galois extension Fa,i of F of degree at most ni, (2) an automorphismσa,i ∈Gal(Fa,i/F),
(3) a finite extension Fa,∗i of F,
(4) an embeddinga,i :Fa,i → Fa,∗i, and,
(5) for each j =0, . . . ,i, an embeddinga,j,i : Fa,j∗ → Fa,i∗ ,
such thatFa,i∗ is the compositum of alla,j,i(a,j(Fa,j)), j i, and, fork j i,a,i,i = idF∗
a,i anda,j,i◦a,k,j =a,k,i.
We intentionally do not keep track of the size of the tuples in order to avoid an overload of notation. We will also handle finite and infinite sequences rather sloppily.
Lemma C.3. For eachλ < ωandn=(ni)i<λthere is anLring-formulaαλ,n such that, for a tuple aof elements of F, F|αλ,n(a)iffa is admissible of lengthλ and degreesn.
Proof. This is clear; see also [6,§1].
Definition C.4. Every x∈ S(Gal(F)) is of the formg N for some open normal subgroup N Gal(F), and we denote byFxthe finite extension ofFthat is the fixed field ofN, and byσx ∈Gal(Fx/F)the automorphism corresponding tox under the natural isomorphism Gal(F)/N ∼=Gal(Fx/F)induced by restriction.
Definition C.5. A sequence x=(xi)i<λ of elements of S(Gal(F)) and an admissible sequenceaof lengthλarecompatibleif there existF-isomorphismsφi :Fx0· · ·Fxi → Fa,∗i withφi(Fxi)=a,i(Fa,i)– soφi induces an isomorphismφ˜i :Fxi → Fa,i – andφ˜i◦σxi = σa,i◦ ˜φi for alli < λ, such that, for j i,φi|Fx0···Fx j =a,j,i◦φj.
Lemma C.6. Let x anda be compatible sequences of lengthλ < ω.
(a) For every xλ∈S(Gal(F)) there exists a such that aa is admissible of lengthλ+1 and(x0, . . . ,xλ)andaa are compatible.
(b) For every a such that aa is admissible of length λ+1 there exists xλ ∈S(Gal(F)) such that (x0, . . . ,xλ)andaa are compatible.
Proof. Letφi :Fx0· · ·Fxi → Fa,∗i fori< λbe as in DefinitionC.5.
(a). Chooseasuch that there are isomorphismsφ:Fxλ → Faa,λandφλ :Fx0· · ·Fxλ → Faa∗,λ, andaa,λ=φλ◦φ−1,aa,i,λ=φλ◦φi−1,σa,λ=φ◦σxλ◦φ−1.
(b). Extend the isomorphism φλ−−11: Fa,λ−∗ 1→ Fx0· · ·Fxλ−1 to an embedding ψ: Faa∗,λ→(Fx0· · ·Fxλ)˜ with ψ◦aa,λ−1,λ=φ−λ−11 , and choose xλ such that Fxλ = ψ(aa,λ(Faa,λ)), andσxλ corresponds toσaa,λunder ψ◦aa,λ.
Definition C.7. We assign to each bounded Lco-formula ϕ(v0, . . . , vλ−1) and tuple n= (ni)i<ω anLring-formulaϕring,n(u), whereu=u0u1. . ., as follows.
(1) Ifϕ isvi vj, then ϕring,n expresses thatαk+1,n(u), wherek=max{i,j}, and that u,i,k(u,i(Fu,i))⊇u,j,k(u,j(Fu,j)), so thatu induces an embeddingFu,j → Fu,i. (2) Ifϕ isvi vj, then ϕring,n expresses thatαk+1,n(u), wherek=max{i,j}, and that
u induces an embeddingFu,j → Fu,i, and, under this embedding,σu,i|Fu,j =σu,j. (3) Ifϕis P(vi1, vi2, vi3), thenϕring,n expresses thatαk+1,n(u), wherek=max{i1,i2,i3}, and that the three embeddings u,iν,k◦u,iν, ν=1,2,3, induce isomorphisms between the Fu,iν, and, under these isomorphisms,σu,i1◦σu,i2 =σu,i3.
(4) Ifϕ isvi =vj, then ϕring,n expresses thatαk+1,n(u), wherek=max{i,j}, and that u induces an isomorphismφ:Fu,i →Fu,j, andφ◦σu,i =σu,j◦φ.
(5) Ifϕ isvi ∈Gn, thenϕring,n expresses thatαi+1,n(u)and that[Fu,i :F]n.
(6) Ifϕ is of the formψ∧η, thenϕring,n isψring,n∧ηring,n, and ifϕ is of the form ¬ψ, thenϕring,n is¬ψring,n.
(7) Ifϕ is of the form(∃vi ∈Gn)(ψ)andλi is minimal such that the free variables in ψ are among v0, . . . , vλ, then, after renumbering the variables, we can assume
that i=λ, and ϕring,n(u)is
(∃u)(αλ+1,n(uu)∧ψring,n(u0. . .uλ−1u)), where n:=(n0, . . . ,nλ−1,n).
IfΣis a ranked set of coformulas in the variables(vi)i<λ, we letn:=nΣ :=(ni)i<λwith ni minimal such that the formula Gni(vi) is in Σ, and let Σring consist of all αi,n, for i < λ, and allϕring,n, forϕ∈Σ.
Lemma C.8. Let x, a be compatible sequences of length λω and degrees n, and let ϕ(v0, . . . , vλ−1) be a bounded Lco-formula. Then S(Gal(F))|ϕ(x) if and only if F | ϕring,n(a).
Proof. This is proven by case distinction according to Definition C.7. In cases (1)–(5), the claim follows immediately from the compatibility. In case (6), the claim is trivial by induction. In case (7), the claim follows by induction and LemmaC.6.
Proposition C.9. LetΣ be a ranked set of coformulas in the variablesv0, v1, . . .. ThenΣ is cosatisfied inGal(F)if and only ifΣring is satisfied in F.
Proof. Letn=nΣ as in Definition C.7.
IfΣis cosatisfied inGal(F)byx=(x0,x1, . . .), then[Fxi : F]ni for alli. By applying Lemma C.6 iteratively we get an admissible sequence a of degrees n such thatx and a are compatible. In particular,F |αi,n(a)for everyi. For everyϕ ∈Σ,S(Gal(F))|ϕ(x) implies that F |ϕring,n(a)by LemmaC.8, so Σring is satisfied in F.
Conversely, if Σring is satisfied by a sequence a, then, since Σring contains all αi,n, a is admissible of degrees n. Lemma C.6 gives a sequence x in S(Gal(F)) such that x anda are compatible. For everyϕ ∈Σ, F |ϕring,n(a)implies that S(Gal(F))|ϕ(x)by LemmaC.8, soΣ is satisfied inF.
Corollary C.10. There is a recursive map ϕ →ϕring from bounded Lco-sentences to Lring-sentences such that, for any field F, S(Gal(F))|ϕ if and only if F |ϕring. Proof. This is the special caseΣ = {ϕ}of PropositionC.9.
Corollary C.11. If F isℵ1-saturated, then Gal(F)isℵ1-cosaturated.
Proof. Let Σ be a countable ranked set of coformulas in the variables v0, v1, . . . with parameters in S(Gal(F)) for which every finite subset is cosatisfied in Gal(F), and let n=nΣ. For a finite subset
Σ0 = {(ϕ1)ring,n, . . . , (ϕm)ring,n, αi1,n, . . . , αil,n} ⊆Σring
chooseλmax{i1, . . . ,il}such that the free variables of theϕi are amongv0, . . . , vλ, and let
Σ0:= {ϕ1, . . . , ϕm,Gn0(v0), . . . ,Gnλ(vλ)}.
Since (ϕi)ring,n=(ϕi)ring,nΣ0, we see that Σ0 ⊆(Σ0)ring, so, since Σ0 is cosatisfied in Gal(F), Proposition C.9 gives that (Σ0)ring, and hence Σ0 is satisfied in F. Therefore, sinceFisℵ1-saturated,Σring is satisfied inF, and henceΣis cosatisfied inGal(F), again by PropositionC.9.
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