ELIMINATING FIELD QUANTIFIERS IN ALGEBRAICALLY MAXIMAL KAPLANSKY FIELDS
arXiv:1707.03188v1 [math.LO] 11 Jul 2017
YATIR HALEVI∗ AND ASSAF HASSON† Abstract. We prove elimination of field quantifiers for algebraically maximal Kaplansky fields.
1. introduction The aim of this note is to give a complete, essentially self-contained, proof that algebraically maximal Kaplansky fields eliminate field quantifiers (in the sense1, e.g., of [8, Definition 1.14]). While probably folklore, we could not find a proof in the literature, though several closely related theorems do exist. In [5, Theorem 2.6] Kuhlmann proves that such valued fields admit quantifier elimination relative to a structure he calls an amc-structure of level 0. It is well known that this structure is essentially the RV-structure (see for instance [2, Section 3.2]). In this language, he proves that if L and F are models of a theory of an algebraically maximal Kaplansky field and K is a common substructure then RVL ≡RVK RVF =⇒ (L, v) ≡(K,v) (F, v), where the valued fields are considered, for instance, in the Ldiv language. This is proved by showing that every embedding RVL ֒→ RVF (over RVK ) lifts to an embedding (L, v) ֒→ (F, v) (over (K, v)), when assuming that F is |L|-saturated, Using this result B´elair proves that, in equi-characteristic (p, p), every algebraically maximal Kaplansky field eliminates field quantifiers in the Denef-Pas language, the 3-sorted language enriched with an angular component map (see [1, Lemma 4.3]). It seems, though it is not claimed, that B´elair’s proof may apply to the mixed characteristic case as well. In what follows we give an alternative proof, by reproducing Kuhlmann’s proof from [5, Section 3] but working entirely in the Denef-Pas language. There is nothing essentially new in the results of this note and the proofs follow closely Khulmann’s (with some ideas borrowed from the proof of the (0, 0) henselian case, for instance from [9, Theorem 5.21]). 2. preliminaries We review some terminology and definitions. For a valued field (K, v) let vK denote the value group, Kv the residue field and res the residue map. Valued fields will be considered in the 3-sorted language, with a sort for the base field, the value group and the residue field, with the obvious functions and relations. Date: today. ∗ The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No. 291111. † Supported by ISF grant No. 181/16. 1 See also [7, Appandix A] for a more detailed discussion. 1
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For a valued field (K, v) an angular component map is a multiplicative group homomorphism ac : K × → Kv × such that ac(a) = res(a) whenever v(a) = 0, we extend it to ac : K → Kv by setting ac(0) = 0. Fact 2.1. [9, Corollary 5.18] Every valued field (K, v) has an elementary extension with an angular component map on it. A valued field, admitting an angular component map will be referred to as an ac-valued field. The 3-sorted language of valued fields together with a function symbol for the ac-map (angular component map), is called the Denef-Pas language. Finally: Definition 2.2. A valued field (K, v) of residue characteristic p > 0 is a Kaplansky field if the value group is p-divisible, the residue field is perfect and the residue field does not admit any finite separable extensions of degree divisible by p. (K, v) is algebraically maximal if if does not admit any immediate algebraic extension. Fact 2.3. [3, Theorem 5] Every Kaplansky field has a unique immediate maximal extension. 3. extending embeddings The following results are proved in [9] for the (0, 0) case. We use results from [4] to give some slight generalizations. Lemma 3.1. [9, Lemma 5.20] Let L, F be ac-valued fields, f : L → F an isomorphism of ac-valued fields g : L′ → F ′ a valued-field isomorphism extending f to some ac-valued field extensions L′ /L and F ′ /F such that vL = vL′ . Then g commutes with the ac-map. × Proof. Let x ∈ L′ , we may write x = x1 x2 where x1 ∈ L and x2 ∈ OL ′ and thus ac(x) = ac(x1 )res(x2 ) and
f (ac(x)) = f (ac(x1 ))f (res(x2 )) = ac(f (x1 ))res(f (x2 )) = ac(f (x)). A field K has p-roots if K p = K. If K is a valued field it is Artin-Schreier closed if every irreducible polynomial of the form xp − x − c has a root in K for p = char(Kv) > 0. Lemma 3.2. Let L1 and L2 be henselian ac-valued fields, K a common henselian ac-valued subfield and f : K → L2 be the embedding (so f commutes with the ac-map). Further assume that if char(Kv) = p > 0 then L1 , L2 are both either Artin-Schreier closed or have p-roots. If vL1 ≡vK vL2 then for any γi ∈ vLi with tp(γ1 /vK) = tp(γ2 /vK) there exist bi ∈ Li with v(bi ) = γi such that f may be lifted to an isomorphism of ac-valued fields f : K(b1 ) → K(b2 ), in particular K(bi ) are ac-valued fields. Proof. By the previous lemma we may, of course, assume that γi ∈ / vK. We now break the proof in two: Case 1: Assume that nγ ∈ / vK for every n ≥ 1 and let bi ∈ Li be any elements × such that with v(bi ) = γi . Replacing bi , if needed, by bi /ui with ui ∈ OL i ac(ui ) = ac(bi ), we have ac(bi ) = 1.
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Because γ is not in the divisible hull of vK, necessarily bi is transcendental over K. By [9, Lemma 3.23] (or [4, Lemma 6.35]), v extends uniquely to K(bi ), K(bi )v = Kv and vK(bi ) = vK ⊕ Zγi . It remains to show that this extension commutes with the ac-map. Let a ∈ K(b1 ). Because vK(b1 ) = vK ⊕ Zγ1 there exists c with v(c) ∈ vK such that v(a) = v(cbn1 ) for some n ∈ Z. Thus a = (cbn1 ) · a/(cbn1 ) and v(a/(cbn1 )) = 0. So ac(a) = ac(cbn1 )res(a/(cbn1 )) = ac(c)res(a/(cbn1 )) and f (ac(a)) = f (ac(c))f (res(a/(cbn1 ))) = ac(f (c)f (bn1 )f (a/(cbn1 )))) = ac(f (a)). Case 2: Assume that qγi ∈ vK for some prime q. Case 2.1: Assume that q 6= char(Kv). Let d ∈ K with v(d) = qγ. We may × replace d by d/u with u ∈ OK such that ac(u) = ac(d) and then ac(d) = 1. Let ai ∈ Li with v(ai ) = γi and ac(ai ) = 1. Then the polynomial Pi (x) = xq −d/aqi is over OLi and satisfies v(P (1)) > 0 and v(P ′ (1)) = 0. Indeed, v(d/aqi ) = 0 and res(d/aqi ) = ac(d)/ac(aqi ) = ¯1, so P¯ (¯ 1) = 0. Since q 6= char(Kv) we automatically get that P¯ is separable, so P¯ ′ (1) 6= 0. This gives ui ∈ Li such that Pi (ui ) = 0 and u¯i = 1. Now let bi = ai ui , clearly bqi = d and ac(bi ) = 1. By [4, Lemma 6.40], K(bi )v = Kv and vK(bi ) = vK ⊕ (Z/qZ)γi . Now f |K(bi ) commutes with the ac-map just as in case 1. Case 2.2: Assume that q = p = char(vK) > 0. Since both L1 and L2 are either Artin-Schreier closed or have p-roots, just as in Case 2.1 we may use [4, Lemma 6.40] to extend the isomorphism (see the statement of [4, Lemma 6.40]). Lemma 3.3. Let L1 and L2 be henselian ac-valued field, K a common henselian ac-valued subfield and f : K → L2 the embedding (so f commutes with the ac-map). Assume that L1 v ≡Kv L2 v and let αi ∈ Li v, with αi separable over Kv if it is algebraic, and tp(α1 /Kv) = tp(α2 /Kv). Then there exist ai ∈ Li with a ¯i = αi such that f may be lifted to an isomorphism f : K(a1 ) → K(a2 ) of ac-valued fields. In particular K(ai ) are ac-valued fields. Proof. It will suffice, by Lemma 3.1, to show that we can extend f to an isomorphism of valued field extensions preserving the value group of K. We break into two cases. Case 1: Assume that αi ∈ Li v are transcendental over Kv. Pick any ai ∈ OLi such that a ¯i = αi . They are necessarily transcendental over K. Either by [4, Lemma 6.35] or by [9, Lemma 3.22], v extends uniquely to K(ai ) and vK(ai ) = vK. Case 2: If αi is algerbaic, let P (X) ∈ OK [x] be monic such that P¯ (x) is the minimal polynomial of αi over Kv. Pick bi ∈ OL such that ¯bi = αi . As the αi are separable over Kv, P¯ is separable and since v(P (bi )) > 0 and v(P ′ (bi )) = 0 we may find ai ∈ OL such that P (ai ) = 0 and a¯i = α. Either by [4, Lemma 6.41] or [9, Lemma 3.21], v extends uniquely to K(ai ) and vK(ai ) = vK. 4. eliminating field quantifiers Let T be the theory of an algebraically maximal Kaplansky field. We consider it in the Denef-Pas language. The proof of our main result follows [5, Section 3]
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step-by-step. Throughout we verify that the embedding constructed respects the angular component map. Recall that a valuation transcendence basis T for an extension L/K is a transcendence basis for L/K of the form T = {xi , yi : i ∈ I, j ∈ J} such that the values vxi , i ∈ I form a maximal system of values in vL which are Q-linearly independent over vK and the residues y¯j , j ∈ J form a transcendence basis of Lv/Kv. Also recall that an algebraic extension of henselian fields L/K is tame if for every finite subextension K ′ /K: (1) K ′ v/Kv is separable. (2) if p = char(Kv) > 0 then (vK ′ : vK) is prime to p. (3) K ′ /K is a defectless extension. K will be called tame if it is henselian and every algebraic extension is a tame field. Recall ([4, Lemma 13.37]) that algebraically maximal Kaplansky fields are tame. Lemma 4.1. Let L and F be henselian ac-valued fields and K a common acvalued field. Assume that L is a tame algebraic extension of the henselization of K. Then for every embedding τ = (τΓ , τk ) : (vL, Lv) ֒→ (vF, F v) over K, there is an embedding of L in F over K which induces τ and commutes with the ac-map. Proof. By Lemma 3.1, the uniqueness of the henselization of K and the fact that it is an immediate extension we may assume that K is henselian (the unique isomorphism of the henselisation of K in L with its henselisation in F respects the ac-map). Since every algebraic extension is the union of its finite sub-extensions we may further assume that L/K is finite. Since Lv/Kv is finite an separable it is simple (recall L/K is tame). Assume that Kv(α) = Lv. By Lemma 3.3(Case 2) there exists c ∈ L with c¯ = α, K(c)v = Lv and vK(c) = vK such that we may extend the embedding of K ֒→ F to an embedding K(c) ֒→ F respecting the ac-map. Set K ′ := K(c). Since L/K ′ is finite, the group vL/vK ′ is a finite torsion group: vL/vK ′ = (γ1 + vK)Z ⊕ · · · ⊕ (γr + vK)Z, and by tameness the order of each γi is prime to p = char(Kv). Using Lemma 3.2(Case 2.1) repeatedly there exist d1 , . . . , dr ∈ L with v(di ) = γi such that L/K ′ (d1 , . . . , dr ) is an immediate extension and we may extend the embedding of K ′ ֒→ F to an embedding K ′ (d1 , . . . , dr ) ֒→ F that preserves the ac-map. Finally, K ′ (d1 , . . . , dr ) is a tame field since it is an algebraic extension of a tame field, thus L/K ′(d1 , . . . , dr ) is defectless and immediate. Since henselian defectless fields are algebraically maximal, it follows that K ′ (d1 , . . . , dr ) = L and we are done. Lemma 4.2. Let L and F by henselian ac-valued field and K a common ac-valued field. Assume that L admits a valuation transcendence basis T such that L is a tame extension of K(T )h . Then for every embedding τ = (τΓ , τk ) : (vL, Lv) ֒→ (vF, F v) over K, there is an embedding of ac-valued fields L ֒→ F over K inducing τ . Proof. By using Lemma 3.2(Case 1) and Lemma 3.3(Case 1) repeatedly we may extend the embedding K ֒→ F to an embedding K(T ) ֒→ F over K respecting the ac-map, and inducing the embedding τ . The result now follows from Lemma 4.1 The next lemma is now key:
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Lemma 4.3. Let K be an algebraically maximal Kaplansky field with char(Kv) = p > 0. Then K contains p-roots, i.e. K p = K. Proof. If char(K) = p then this follows from the fact that K is perfect ([6, Corrolaries 3.3, 3.4]). If char(K) = 0 Kaplansky [3, Lemma 15] proves this for maximal Kaplansky fields. He shows it by constructing a pc-sequence of algebraic type. The same proof follows through verbatim for algebraically maximal Kaplansky fields. Alternatively, one may use Kaplansky’s result for maximal fields and then apply [6, Theorem 7.1]. Fact 4.4. [5, Lemma 3.12] Let L and F be two algebraically maximal Kaplansky fields (and hence defectless) and K a common henselian subfield. Assume that both vL/vK and vF/vK are torsion groups and both Lv/Kv and F v/Kv are algebraic extensions. If K does not admit any non-trivial tame algebraic extension inside L or F , then the relative algebraic closures of K in L and F are isomorphic. The following embedding theorem is, as usual, the main result: Theorem 4.5. Let L, F |= T with F |L|+ -saturated and K a common subfield, all with angular component maps and in the Denef-Pas language. Then for every embedding τ = (τΓ , τk ) : (vL, Lv) ֒→ (vF, F v) over K, there is an embedding of L in F over K which induces τ and preserve s the ac-map. Proof. In ordered to embed L in F (over K) it is suffices to embed every finitely generated sub-extension. So we may assume that L/K is a finitely generated extension. Let T = {xi , yi : i ∈ I, j ∈ J} be such that {vxi : i ∈ I} is a maximal system of Q-linearly independent values in vL over vK and the residues {¯ yj : j ∈ J} are a transcendence basis of Lv/Kv. T is algebraically independent over K by [4, Lemma 6.30]. Let L′ be the maximal tame algebraic extension of the henselization K(T )h in L. By definition it is an algebraic extension so T is a transcendence basis for L′ /K and it is a tame extension of K(T )h . We may use Lemma 4.2 and embed L′ in F over K, this embedding induces τ and commutes with the ac-map. Since L′ is the maximal tame algebraic extension of K(T )h , necessarily vL/vL′ is a p-group and Lv/L′ v is a purely inseparable algebraic extension. By Fact 4.4, we may extend the embedding to L′′ , its relative algebraic closure in L. By [4, 13.40] L/L′′ is immediate (use the maximality of T ) and thus vL′′ /vL′ is a pgroup, L′′ v/L′ v is a purely inseparable algebraic extension and L′′ is an ac-valued algebraically maximal Kaplansky field. We must show that the embedding L′′ ֒→ F preserves the ac-map. By Lemma 4.3, L′′ contains p-roots and hence by repeatedly applying Lemma 3.2(Case 2.2) we may extend the embedding of L′ ֒→ F to the intermediary ac-valued field L′ ⊆ L′′′ ⊆ L′′ with vL′′′ = vL′′ , in such a way that it commutes with the ac-map. By uniqueness of the extension of the valuation this agrees with the embedding L′′ ֒→ F . Now use Lemma 3.1. Recall that L′′ is also an algebraically maximal Kaplansky ac-valued field and L/L′′ is immediate. By the uniqueness of the maximal immediate extension of L′′ (and of L) we may extend the embedding to L, it preserves the ac-map by Lemma 3.1. Corollary 4.6. Let L, F |= T and K a common substructure. If (vL, Lv) ≡(vK,Kv) (vF, F v)
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then L ≡K F. As a result, T eliminates field quantifiers and the value group and residue field are stably embedded as pure structures. Proof. There exists elementary extension L∗ and F ∗ over L and F such that vL∗ ∼ =vK vF ∗ and L∗ v ∼ =Kv F ∗ v. Since L∗ ≡K F ∗ implies L ≡K F , we may assume from the start that τ = (τΓ , τk ) : (vL, Lv) ∼ =(vK,Kv) (vF, F v). The rest is standard and follows from the theorem. References [1] Luc B´ elair. Types dans les corps valu´ es munis d’applications coefficients. Illinois J. Math., 43(2):410–425, 1999. [2] Joseph Doyle Flenner. The relative structure of henselian valued fields. ProQuest LLC, Ann Arbor, MI, 2008. Thesis (Ph.D.)–University of California, Berkeley. [3] Irving Kaplansky. Maximal fields with valuations. Duke Math. J., 9:303–321, 1942. [4] Frantz-Victor Kuhlmann. Valuation theory. Available on http://math.usask.ca/˜fvk/Fvkbook.htm, 2011. [5] Franz-Viktor Kuhlmann. Quantifier elimination for Henselian fields relative to additive and multiplicative congruences. Israel J. Math., 85(1-3):277–306, 1994. [6] Franz-Viktor Kuhlmann. The algebra and model theory of tame valued fields. J. Reine Angew. Math., 719:1–43, 2016. [7] Silvain Rideau. Some properties of analytic difference valued fields. J. Inst. Math. Jussieu, 16(3):447–499, 2017. [8] Saharon Shelah. Dependent first order theories, continued. Israel J. Math., 173:1–60, 2009. [9] Lou van den Dries. Lectures on the model theory of valued fields. In Model theory in algebra, analysis and arithmetic, volume 2111 of Lecture Notes in Math., pages 55–157. Springer, Heidelberg, 2014. ∗ Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, Jerusalem 91904, Israel, E-mail address:
[email protected] URL: http://ma.huji.ac.il/~yatirh/ † Department of mathematics, Ben Gurion University of the Negev, Be’er Sehva, Israel E-mail address:
[email protected] URL: http://www.math.bgu.ac.il/~hasson/
