Transcendental extensions of a valuation domain of rank one

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1 accepted for publicatio i the Proc. Amer. Math. Soc. (2016). Trascedetal extesios of a valuatio domai of rak oe Giulio Perugielli 11 May 2016 Abstract Let V be a valuatio domai of rak oe ad quotiet field K. Let K be a fixed algebraic closure of the v-adic completio K of K ad let V be the itegral closure of V i K. We describe a relevat class of valuatio domais W of the field of ratioal fuctios K(X) which lie over V, which are idexed by the elemets α K { }, amely, the valuatio domais W = W α = {ϕ K(X) ϕ(α) V }. If V is discrete ad π V is a uiformizer, the a valuatio domai W of K(X) is of this form if ad oly if the residue field degree [W/M : V/P ] is fiite ad πw = M e, for some e 1, where M is the maximal ideal of W. I geeral, for α, β K we have W α = W β if ad oly if α ad β are cojugated over K. Fially, we show that the set P irr of irreducible polyomials over K edowed with a ultrametric distace itroduced by Kraser is homeomorphic to the space {W α α K} edowed with the Zariski topology. Keywords: Discrete valuatio domai, Valuatio overrigs, Iteger-valued polyomial, Prüfer domai, Zariski topology, Ultrametric space. MSC codes: 16W60, 13J10, 13B25, 13F20 1 Itroductio Let V be a valuatio domai of rak 1, quotiet field K ad let v be the associated valuatio. Let V ad K be the v-adic completios of V ad K, respectively. Give a field extesio K F ad a valuatio domai W of F, we say that W lies above V if W K = V. If F = K(θ) is a simple separable algebraic extesio of K ad p K[X] is the miimal polyomial of θ, the the valuatio domais W of F which lie above V are well-kow: they are rak 1 valuatio domais which are i oe-to-oe correspodece with the irreducible factors over K of p(x) (see for example [2, Chapter VI,. 8, 2., Propositio 2 ad Corollaire 2] or [15, Chapter 6, B.]). More precisely, there exists a fiite set of elemets {θ 1,..., θ } i a fixed algebraic closure K of K (i.e., the roots of p(x) i K) such that the above valuatio domais W are equal to W θi = {g(θ) K(θ) g(θ i ) is itegral over V }, for i = 1,..., ; moreover, W θi = W θj if ad oly if θ i, θ j are cojugated over K (i.e., θ i, θ j are roots of the same irreducible factor of p(x) over K). If istead we cosider a simple trascedetal extesio K(X) of K, the structure of the set of valuatio domais of K(X) which lie above V is much richer (see for example [1, 8, 9, 11, 12, 13, 16]). To begi with, it is well-kow that the rak of W is 1 or 2 (see for example [2, Chapt. VI, 10, 1

2 Corollaire 1, p. 162]). The aim of this paper is to give a explicit descriptio of a particular class of these valuatio domais, which, likewise the previous algebraic case, arise from the elemets α K { }, amely W = W α = {ϕ K(X) ϕ(α) is itegral over V }. The descriptio of these valuatio domais is accomplished i Propositio 2.2. Whe V is a discrete valuatio domai of rak oe, we give i Theorem 2.5 sufficiet coditios o a valuatio domai W of K(X) lyig over V to be of the form W α, for some α K { }, amely: i) the residue field degree [W/M : V/P ] is fiite; ii) πw = M e, for some e 1; where π is a uiformizer of V ad M is the maximal ideal of W. I cotrast with the fiite algebraic case recalled above, where the extesios of V to K(θ) are give by a fiite set of elemets i K, i the trascedetal case oe has to cosider all the ucoutably may elemets of K i order to obtai all the valuatio domais W K(X) of the above form. As i the algebraic case, for α, α K, W α = W α if ad oly if α ad α are cojugate over K (Theorem 3.2). Moreover, it turs out that these valuatio domais are precisely the uitary valuatio overrigs of a class of geeralized iteger-valued polyomial rigs which was itroduced i [10]: give a fiite field extesio F of K, let V F be the itegral closure of V i F. We set It K (V F ) = {f K[X] f(v F ) V F } (1.1) Note that It K (V F ) is the cotractio to K[X] of It(V F ) = {f F [X] f(v F ) V F }, the classical rig of iteger-valued polyomials over V F. Give a valuatio domai W K(X) as above, we show that there exists a fiite extesio F of K such that W is a overrig of It K (V F ). I the secod sectio we give the characterizatio of the aforemetioed valuatio domais W of K(X). We show how the valuatio domais W α, for α K, are related to the work of Kaplasky about immediate extesios of a valued field i [6], see Remark 2.6. More geerally, whe α rages i K, we show the coectio with the work of MacLae i [11, 12] about approximatios of trascedetal extesios of a DVR, see Remark 2.7. The valuatio domais W α, α K, appear also i the recet paper [13], which deals with extesios of a DVR to a trascedetal extesio of its field of quotiets i the spirit of MacLae. Furthermore, as a applicatio, we show that the Prüfer domais of polyomials betwee Z[X] ad Q[X] costructed i [10] ca be represeted as rigs of iteger-valued polyomials (Remark 2.8). I the third sectio, for a geeral rak oe valuatio domai V, we show that the set W = {W α α K} is i oe-to-oe correspodece with the followig sets: the set P irr of irreducible polyomials over K; the set W K[X] = {W α K[X] α K} (Theorem 3.2). I particular, this allows us to reduce may cosideratios to polyomials rather tha to ratioal fuctios. Moreover, if we edow P irr with the ultrametric distace (p, q) = mi{ α β α, β K, p(α) = q(β) = 0} ad W ad W K[X] with the Zariski topology, these three spaces are homeomorphic (Theorem 3.4). A first evidece of this result is cotaied i a paper of Gilmer, Heizer, Latz ad Smith, where, for a DVR V with fiite residue field, it is proved that the uitary maximal spectrum of the rig It(V ) (that is, those maximal ideals whose cotractio to V is equal to the maximal ideal of V ) is homeomorphic to V (see [5, p. 677]). Sice V is homeomorphic to the space of moic liear polyomials edowed with the above distace (, ), Theorem 3.4 is a geeralizatio of this result. 2

3 2 Mai result Throughout the paper, we adopt the followig otatio. Let V be a valuatio domai of rak 1 with quotiet field K. We deote by v the associated valuatio o K ad by P, V, K the v-adic completio of P, V, K, respectively. Note that V K = V, V is a valuatio domai of rak 1 of K with residue field isomorphic to the residue field of V ad v exteds uiquely to the valuatio of K associated to V, which we still deote by v. Let K be a fixed algebraic closure of K ad V the itegral closure of V i K. It is well kow that v admits a uique extesio to K ([15, Chapt. 5, A.]), which agai we deote by v ad whose valuatio rig is V, thus V = {α K v(α) 0}. For α K, we deote by V α the valuatio domai of rak 1 of the fiite field extesio K(α) of K, which is equal to the itegral closure of V i K(α), ad by P α the maximal ideal of V α. We deote by v α the valuatio of K(α) associated to V α (thus, the restrictio of v to K(α)). We recall the otio of Gaussia extesio of v. Give f(x) = i 0 a ix i K[X], we set v G (f) mi{v(a i ) i = 0,..., }, ad this fuctio exteds i the atural way to a valuatio of K(X), called the Gaussia extesio of v. The valuatio domai of the Gaussia extesio is equal to V [X] P [X] (see for example [4, Propositio 18.7] or [2, Chapt. VI, 10]) ad is the uique extesio of V to K(X) such that X is trascedetal over the residue field ([2, Chapt. VI, 10, Prop. 2]). Give a field extesio K F, a valuatio domai W of F is immediate over K if the value groups ad the residue fields of W ad W K are the same, respectively. For α K, ote that the elemets of the value group of V α K(α) are of the form v α (g(α)), where g K[X]. If h K[X] is such that v G (h g) is sufficietly greater tha v α (g(α)), the v α (g(α) h(α)) > v α (g(α)), so that V α is immediate over K(α) (see [2, 10, Exercise 2, p. 193]). I order to describe all the possible valuatio domais of K(X) we are iterested i, we eed to cosider the projective lie over K, that is, P 1 ( K) K { }. Give a ratioal fuctio ϕ K(X) ad α K, ϕ(α) is a elemet of P 1 ( K); we say that ϕ is ot defied at α if ϕ(α) =. We also set ϕ( ) ψ(0), where ψ(x) = ϕ(1/x), so that each ratioal fuctio o K determies a map from P 1 ( K) to itself, which is cotiuous with respect to the v-adic topology (see also Remark 3.1). We itroduce ow the followig defiitio. Defiitio 2.1. Let α P 1 ( K). We cosider the set of ratioal fuctios ϕ(x) over K which are defied at α ad such that their evaluatio at α is itegral over V : W α {ϕ K(X) ϕ(α) V }. Clearly, for α P 1 ( K), ϕ W α ϕ(α) V α v(ϕ(α)) 0, where by covetio we set V = V ad P = P. We also set K( ) K. The followig propositio characterizes W α. It is a stadard result, but for the sake of the reader we give a proof here. Propositio 2.2. Let α P 1 ( K). The W α is a valuatio domai of K(X) which lies over V with maximal ideal M α = {ϕ K(X) v(ϕ(α)) > 0} ad rak 1 or 2. The rak of W α is 1 if ad oly if α is i K ad is trascedetal over K, it is 2 if either α K is algebraic over K or α =. If the rak of W α is 2, α K ad q K[X] is the miimal polyomial of α over K, the the DVR K[X] (q) is the valuatio overrig of W α. If α =, the K[ 1 X ] ( 1 X ) is the valuatio 3

4 overrig of W. Moreover, the residue field of W α is isomorphic to the residue field of V α ad the value group of W α /Q α is also isomorphic to the value group of V α, where Q α = 0 if W α has rak 1 ad Q α is the height oe prime of W α if W α has rak 2. Proof. Let α be ay give elemet of P 1 ( K). It is straightforward to show that W α is a valuatio domai of K(X) which lies over V ad with maximal ideal M α = {ϕ K(X) v(ϕ(α)) > 0}. I fact, give a ratioal fuctio ϕ K(X) defied at α, that is, ϕ(α) K(α), either ϕ(α) V α or ϕ(α) 1 V α, sice V α is a valuatio domai of K(α). Both of these coditios hold if ad oly if ϕ W α \ M α, so the latter is the multiplicative group of uits of the valuatio domai W α (ad so M α is its maximal ideal). We remark that W α ca be realized as the pullback of the valuatio domai of rak 1 V α of K(α), via the evaluatio homomorphism ev α at α: ev α : K(X) K(α) { } { ϕ(α), if ϕ(x) is defied at α ϕ(x), otherwise (2.1) The image of ev α i (2.1) is cotaied i K(α) if ad oly if α is trascedetal over K if ad oly if the kerel Q α = {ϕ W α ϕ(α) = 0} of the restrictio (ev α ) Wα : W α V α is equal to (0). If α is algebraic over K, the the domai of defiitio of ev α is the DVR K[X] (q), where q K[X] is the miimal polyomial of α over K, so ev α (K[X] (q) ) = K[X]/(q) = K(α). Moreover, sice Q α V = (0), the height oe prime ideal Q α of W α is equal to q(x)k[x] (q) i the algebraic case ad the oe-dimesioal valuatio overrig of W α is equal to K[X] (q). Suppose α K. If α is trascedetal over K, the (ev α ) Wα W α V α is a ijective homomorphism that exteds to a homomorphism of quotiet fields ev α : K(X) K(α). Thus v α ev α defies a valuatio of real rak 1 o K(X) whose associated valuatio rig is W α, hece W α is a valuatio domai of rak 1 of K(X). Reciprocally, if W α is a valuatio domai of rak 1 of K(X) ad Q α 0 (the kerel of (ev α ) Wα ), the Q α = M α ad Q α V (0), which is a cotradictio. Therefore, Q α = 0 ad α is trascedetal over K. Hece, for α K, W α is a valuatio domai of rak 1 if ad oly if α is trascedetal over K ad it is a valuatio domai of rak 2 otherwise. We prove ow the last claims. Sice the image of W α via ev α is equal to the valuatio domai V α K(α) ad V α is immediate over K(α), it follows that the residue field of V α is isomorphic to the residue field of W α. Moreover, sice the kerel of (ev α ) Wα is Q α, the also the value group of V α is isomorphic to the value group of W α /Q α. The case α = is treated by cosiderig the chage of variable K(X) K(Y ), X Y 1 X, so W K(X) is easily see to correspod to W 0 i K(Y ). For the rest of this sectio, we adopt the followig assumptios ad otatios: let V be a discrete valuatio domai of rak 1 (DVR) ad let π V be a uiformizer of V, that is, π is a geerator of the maximal ideal P of V. Note that V ad V α are also DVRs, V is a o-discrete valuatio domai of rak 1 ad that π is also a uiformizer of V. O the other had, recall that a discrete valuatio domai is a valuatio domai whose value group is discrete (ot ecessarily of rak 1, see [17, Chapter VI, (A) p.48]). Thus, by Propositio 2.2, W α is a discrete valuatio domai of K(X) of rak 1 or 2 (otice that W α /Q α is a DVR, see also Remark 2.3) ad the residue field of W α is a fiite extesio of the residue field of V. I 4

5 additio, πw α = Mα, e where e is the ramificatio idex of P α over P. I fact, sice we clearly have Mα = {ϕ K(X) ϕ(α) P α } for each 1, it follows that π V α = P α e π P α e e+1 \ P α π Mα e \ M e+1 α. Remark 2.3. By Propositio 2.2 ad the Hasse Existece Theorem ([15, Chapter 6, Theorem 4]), for each pair of positive itegers e, f 1, there exists a valuatio domai W = W α of K(X) lyig over V, where α K, whose residue field has degree f over V/P ad πw = M e. I fact, by the aforemetioed result of Hasse, there exists a algebraic separable extesio K(α) of K with ramificatio idex e ad residue field degree f, where α K (i.e., a primitive elemet). Now, W α is the pullback of V α via the evaluatio morphism ev α ad we ca apply Propositio 2.2 to get that W α has residue field degree equal to f ad πw α = M e α. Give α K, it is ot difficult to give a explicit represetatio of the associated valuatio w α : K(X) Γ Wα, where Γ Wα is the correspodig value group. If the rak of W α is 1 the, as we saw i Propositio 2.2, V α is immediate over K(X) (via the embeddig ev α ), so: w α (ϕ) = v α (ϕ(α)), ϕ K(X) I particular, ote that Γ Wα = Γ Vα, the value group of V α. Suppose ow the rak of W α is 2 ad let q K[X] be the miimal polyomial of α. By [17, Chapt. VI, 10, Thm. 17 & p. 48], the value group of W α is order-isomorphic to Γ q Γ Wα/Q α, where Q α is the height oe prime ideal of W α, Γ q is the value group of the DVR (W α ) Qα = K[X] (q) ad Γ Wα/Q α is the value group of W α /Q α. By the proof of Propositio 2.2, V α cotais W α /Q α ad is immediate over it, so, i particular, Γ Wα/Q α = Γ Vα. Give ϕ K(X) there exist k Z ad g, h K[X], coprime with q(x), such that ϕ(x) = q(x) k g(x) h(x). The ( ) g(α) w α (ϕ(x)) = (k, v α ) Γ q Γ Wα/Q h(α) α Uder the curret assumptio that V is a DVR, we show i Theorem 2.5 that the valuatio domais of Propositio 2.2 are the oly valuatio domais W of K(X) lyig over V whose residue field degree (over V/P ) is fiite ad such that πw = M e, for some e 1. Moreover, the valuatio domais W α, α V, are precisely the uitary valuatio overrigs of a particular class of rigs of iteger-valued polyomials which we ow recall (a valuatio domai W of K(X) lyig over V is called uitary if its ceter o V is the maximal ideal P ). As i the itroductio, for a fiite field extesio F of K, we deote by V F the itegral closure of V i F, which is a Dedekid domai. Give a o-zero prime ideal P of V F, which ecessarily lie over P, we deote by V F,P the localizatio of V F at P. We defie also the followig rig of iteger-valued polyomials: It K (V F,P ) {f K[X] f(v F,P ) V F,P } Let V F,P be the completio of V F,P. We will use the well-kow fact that It K (V F,P ) = It K ( V F,P ) = {f K[X] f( V F,P ) V F,P } (2.2) which is based o the cotiuity of the polyomials with respect to the v-adic topology. 5

6 Before givig the mai result of this sectio, we eed the followig result, which may be wellkow, but for the sake of reader we give a complete proof, which is a adaptio of the argumet give i [15, Chapter 6, Theorem 1, p. 151]. Lemma 2.4. Let V W be complete DVRs with quotiet fields K F ad maximal ideals P, M, respectively. Suppose that the residue field degree [W/M : V/P ] is fiite, equal to a positive iteger f, ad let e be the ramificatio idex of W over V. The [F : K] = ef (so, i particular, F/K is a fiite extesio). Proof. Let π, λ be uiformizers of V ad W, respectively. If e is the ramificatio idex of W over V (which is fiite, sice both valuatio domais are DVRs), the π = λ e u, for some u W. Let y 1,..., y f W be such that their residues modulo M form a V/P -basis of W/M. It is well-kow that the elemets of the set {λ r y j r = 0,..., e 1, j = 1,..., f} are liearly idepedet over K (for example, see first part of the proofs of [2, Chapt VI, 8, 1., Lemma 2] or [15, Chapter 4, F., p. 114] (or also 13.9 of Edler s book; ote that for this result we do t eed the completeess assumptio). For each Z, we set = qe + r, for some q Z ad 0 r < e. We set σ = π q λ r W ; ote that σ has value, for each Z. Let ow z F. There exist 0 Z = Γ W ad u 0 W such that z = σ 0 u 0. Sice u 0 is a uit, there exist a 1,0,..., a f,0 V such that u 0 f j=1 a j,0y j M. Hece, z 1 = z ( f j=1 a j,0y j )σ 0 has value strictly greater tha 0, so we may write z = ( f a j,0 y j )σ 0 + σ 1 u 1 j=1 for some 1 > 0 ad u 1 W. If we cotiue i this way, takig ito accout that F is M-adically complete, we have the followig represetatio for z as a coverget power series: Usig the defiitio of σ, we have that z = z = N( 0 r<e 1 j f f a j, y j )σ j=1 ( q N a j, π q )λ r y j (2.3) Sice K is P -adically complete, for each j the series q N a j,π q is coverget, thus a elemet of K. Hece, (2.3) shows that the elemets λ r y j, for 0 r < e ad 1 j f, are a basis of F over K. I particular, the above Lemma shows that if K is complete, the there are o DVRs i K(X) above V which have fiite residue field degree, i cotrast with the case of whe K is ot complete: if α K is trascedetal over K, the W α is a DVR of K(X) above V which has fiite residue field degree (Propositio 2.2). Theorem 2.5. Let W be a valuatio domai of K(X) with maximal ideal M, such that W lies above V. Suppose that 6

7 i) [W/M : V/P ] = f, ii) πw = M e, for some f, e 1. The there exists α P 1 ( K) such that K(α)/ K has ramificatio idex e ad residue field degree f ad W = W α. I particular, X W α V W is a valuatio overrig of It K (V F,P ), for some fiite field extesio F of K ad prime ideal P V F. I particular, ote that α has degree e f over K ([15, Chapter 6, Theorem 1]). Also, i the case W = W we ecessarily have e = f = 1 (see also Propositio 2.2). The existece of such a valuatio domai W is guarateed also by a more geeral theorem give by Kuhlma [8, Theorem 1.4], but i the preset cotext we have a explicit descriptio of such valuatio domais, see Remark 2.3. Proof. Because of coditio ii), the ideal M is ot idempotet, so M = ϕw, for some ϕ W. Moreover, the rak of W is 1 or 2; we distiguish ow the two cases. Suppose first that W has rak 1, so that W is a DVR of K(X). We cosider the completio Ŵ of W with respect to the M-adic topology. It is well-kow that Ŵ is a DVR with field of fractios K(X), the completio of K(X) with respect to the M-adic topology, ad ϕ is a uiformizer of Ŵ. Sice W lies over V, it follows that K embeds ito K(X) ad Ŵ lies over V. The residue field of Ŵ is isomorphic to the residue field of W ([2, Chapt. VI, 5,. 3, Propositio 5]) ad ϕ e Ŵ = πŵ, sice MŴ is the maximal ideal of Ŵ. I particular, the same assumptios i) ad ii) above for W hold for its completio Ŵ, so the ramificatio idex e(ŵ V ) is equal to e ad the residue field degree f(ŵ V ) is equal to f. By Lemma 2.4, K K(X) is a fiite extesio of degree ef. Therefore, we have a K-embeddig Φ : K(X) K(X) such that α = Φ(X) is algebraic over K, so, without loss of geerality, we may cosider α as a elemet of K; ote that the embeddig Φ is othig else that the evaluatio of X at α. It follows that K(α) is a fiite extesio of K, hece complete, ad sice via the K-embeddig Φ we have the cotaimets K(X) K(α) K(X), it follows that K(α) is equal to the completio K(X) of K(X). Note that Ŵ is isomorphic to the local rig V α of K(α), so via the embeddig Φ we have: W = {ψ K(X) Φ(ψ(X)) = ψ(α) V α } = W α Sice W has rak 1 it follows that α is trascedetal over K, by Propositio 2.2. Suppose ow that W K(X) is a discrete valuatio domai of rak 2. It is kow that the height oe prime ideal Q of W is equal to N M ([4, Theorem 17.3]). Let π Q : W W/Q be the caoical residue map. Sice Q V caot be equal to P because of coditio ii) (i.e., π / M e+1 ), we have Q V = (0). Hece, the restrictio of π Q to V is the idetity, so V W/Q ad K is cotaied i the quotiet field F of the valuatio domai W/Q; i particular, W/Q lies over V. Moreover, W/Q is DVR with maximal ideal M/Q, which is geerated by the residue class of ϕ modulo Q. The localizatio W Q is a DVR of K(X) with maximal ideal Q ad the residue map W Q W Q /Q coicides with π Q over W (ote that the two homomorphisms have the same kerel). Now, X 1 Q W Q = K[X 1 ] (X 1 ), ad i this case the homomorphism W Q W Q /Q = K = F is easily see to be ev, the evaluatio of X at. If istead X W Q, the W Q = K[X] (q), for some irreducible polyomial q K[X]. Therefore, the residue map π Q is 7

8 equal to the restrictio to W of the evaluatio map ev α : K[X] (q) K[X] (q) /Q = K(α), where α is the residue class of X modulo Q, ad K(α) = F is a fiite extesio of K. I either case (X 1 Q α = or X W Q ), sice W/Q is a DVR of F cotaiig V F, it follows that W/Q is equal to V F,P, for some prime ideal P V F (if α =, the V F,P = V ). We idetify α F with its image i the P-adic completio F P = K(α) K of F (see [15, Chapt. 4, L., p. 121]); i this way α is uiquely associated to a elemet of P 1 ( K), which is exactly i the case X 1 Q. Sice V F,P = V F ad α F, we have: W = {ψ K(X) ψ(α) V F,P } = W α. I fact, the value at α of ψ K(X) is i V F,P = W/Q = π Q (W ) if ad oly if ψ is i W, sice π Q = (ev α ) W ad the homomorphisms ev α ad π Q share the same kerel, amely the ideal Q W. Note that the completio of V F,P is isomorphic to V α. Sice W/M = (W/Q)/(M/Q), which is the residue field of W/Q = V F,P, the residue field degree of K(α) over K is f. From the assumptio πw = M e, it follows that π W/Q = (M/Q) e, so the ramificatio idex of K(α)/ K is e. We prove ow the last equivaleces, whether W = W α has rak 1 or 2 (ote that i these cases α ). Suppose that X W. If W α has rak 1, the Φ(X) = α Ŵ = V α, so α is itegral over V. If W α has rak 2, the π Q (X) = α W/Q = V F,P, ad so α is itegral over V (i this case, α is ot ecessarily itegral over V, which is the case exactly whe there is just oe prime ideal P i V F above P ). Suppose that α is itegral over V. I case W = W α has rak 2, the α V F = V F,P, ad so It K (V F,P ) W α. Suppose ow that W α has rak 1 ad let q(x) be the miimal polyomial of α over K, of degree ef. By Kraser s Lemma ([15, Chapt. 5, G.]), if q K[X] is a moic polyomial of degree ef which is v-adically sufficietly close to q(x), the q(x) is irreducible over K, hece also over K. Let F = K(β) be the umber field of degree ef geerated by a root β of q(x). Note that there exists oly oe prime ideal P i V F above P (precisely because q(x) is irreducible over the completio K, see [15, Chapt. 6, B.]); i particular, V F is a DVR ad the P-adic completio of F is isomorphic to K(α) = K(β). Moreover, e(p P ) = e ad f(p P ) = f ([2, Chapt. VI, 5,. 3, Propositio 5]). Let W α = {ψ F (X) ψ(α) V α } be the valuatio domai of F (X) correspodig to α. Sice the P-adic completio of V F is equal to V α, by (2.2) we have It(V F ) = It F ( V α ), ad the latter rig is clearly cotaied i W α, because by assumptio α V α V α. Cotractig everythig dow to K(X) we get It K (V F ) W α. Fially, if W α cotais It K (V F,P ) V [X], the X is i W α. The proof of the last claim of the statemet is ow complete. Remark 2.6. Let W α be a valuatio domai of K(X), with α K. We have see i the proof of Theorem 2.5 that the completio of W α with respect to the M α -adic topology is isomorphic to V α : i fact, if α is trascedetal over K, this is clear. If α is algebraic over K, the the M α -adic completio of W α is equal to the (M α /Q α )-adic completio of W α /Q α, where Q α is the height oe prime ideal of W α, sice Q α = N M α. It follows that if the rak of W α is 1, the W α is immediate over K if ad oly if e = f = 1, thus α K, so α is the limit of a (pseudo-)cauchy sequece {α k } k N K of trascedetal type, accordig to the termiology used by Kaplasky i his paper (see [6, Theorem 2]). Similarly, if the rak of W α is 2, the W α /Q α is immediate over 8

9 K if ad oly if e = f = 1, thus α K. I this latter case, the correspodig Cauchy sequece {α k } k N is of algebraic type. Remark 2.7. We show ow how the valuatio domais W α K(X), α P 1 ( K), are related with the work of MacLae o valuatios of the ratioal fuctio field K(X) which exted a give DVR V of K (see [11, 12]). The class of valuatio domais of Defiitio 2.1 are exactly the costat degree limit valuatios cosidered by MacLae i [11, 7. p. 375] (we refer to that paper for all the uexplaied termiology that follows). Such a valuatio domai is obtaied as a suitable limit of the so-called iductive commesurable valuatios where the degree of the associated sequece of key polyomials is bouded. These kid of valuatio domais ca be of two types, fiite limit valuatios or ifiite limit valuatios (see [11, 6 & 7, pp , & Theorem 7.1] ad [9, Note, p. 108]). The first type of valuatio domais are DVRs of K(X) with residue field which is a fiite extesio of V/P ([11, Theorem 7.1 & Theorem 14.1]), so by Theorem 2.5 ad Propositio 2.2 they correspod to the W α s where α K is trascedetal over K. The secod type of valuatio domai is treated by MacLae as a oe-dimesioal valuatio domai with the value group exteded by addig (they are also called pseudo-valuatios i [13]). I fact, as oted i [9, Lemma 1.23 & p. 109], these last kid of limit valuatios are 2-dimesioal discrete valuatio domais. Moreover, the oe-dimesioal valuatio overrig of such a valuatio domai W is of the form K[X] (q), for some irreducible polyomial q K[X] ad the residue field of W is a fiite extesio of the residue field V/P (this ca be verified directly or also by a suitable modificatio of the origial proof by MacLae i the case of fiite limit valuatios, [11, Theorem 14.1]). Therefore, these valuatios W are exactly those of the form W α, for α K which is algebraic over K. Coversely, suppose we have a valuatio domai W α, α K. By [11, Theorem 8.1] W α ca be realized as a limit valuatio sice its residue field is fiite algebraic over V/P ad the residue field of a iductive commesurable valuatio is a trascedetal extesio of V/P ([11, Theorem 12.1]). Moreover, by [11, Theorem 14.1] the degree of the key polyomials is ecessarily bouded, so W α ca be realized as a costat degree limit valuatio. Remark 2.8. I [10] Loper ad Werer costruct Prüfer domais cotaied betwee Z[X] ad Q[X] by cosiderig arbitrary itersectios of suitable valuatio domais of Q(X), i order to obtai examples of Prüfer domais properly cotaied i It(Z) = {f Q[X] f(z) Z}, the classical rig of iteger-valued polyomials over Z (see [10, Costructio 2.3 & Corollary 2.12]). The valuatio domais used i that costructio are exactly those itroduced i Defiitio 2.1 ad we show ow that the Prüfer domais of [10] ca be represeted as rigs of iteger-valued polyomials. Ideed, for each prime p Z, the valuatio domais V i of [10, Costructio 2.3] have fiite residue field of cardiality bouded by a prescribed positive iteger f p ad maximal ideal M i such that pv i = M ei i, for some e i bouded by a prescribed positive iteger e p. Hece, by Theorem 2.5, each of these valuatio domais is equal to W p,α = {ϕ Q(X) ϕ(α) Z p }, for some α i the absolute itegral closure Z p of the rig Z p of p-adic itegers, whose degree over Q p is bouded by p = e p f p. Let Ω p Z p be the set of all such elemets α. The we have D p W p,α Q[X] = It Q (Ω p, Z p ) = {f Q[X] f(ω p ) Z p } α Ω p which is the rig of polyomials with ratioal coefficiets which are iteger-valued over the set Ω p with respect to Z p. The rig D obtaied i [10, Costructio 2.3] as the itersectio of all the 9

10 rigs {D p p Z prime}, is thus represeted as a itersectio of such rigs of iteger-valued polyomials over differet subsets of itegral elemets over Z p of bouded degree, as p rages through the primes of Z. Followig the otatio of [3], we ca give a more cocise represetatio of D. Let Ω Ω p Z p Ẑ p P p P the D = It Q (Ω, Ẑ) = {f Q[X] f(α) Ẑ, α Ω} (2.4) where, for f Q[X] ad α = (α p ) p Ẑ, we set f(α) (f(α p)) p p P Q p. 3 A ultrametric space of valuatio domais of K(X) I this sectio, we recover our iitial assumptios, thus V is a valuatio domai of rak 1. Throughout this sectio, we deote by W the set of all valuatio domais W α of K(X), as α rages i K. For each α K, we also set: W α {ψ K(X) ψ(α) V } By Propositio 2.2, W α is a valuatio domai of K(X) of rak 2, sice α is algebraic over K by defiitio. Clearly, W α K(X) = W α, ad W α = W α if K is v-adically complete. Moreover, ote that W α is immediate over K(X) if ad oly if α is algebraic over K (if α is trascedetal over K, the W α has rak 1, by Propositio 2.2). We deote by Ŵ the set of all valuatio domais W α of K(X), for α K. We also deote by P irr = P irr ( K) the set of the moic irreducible polyomials over K. Give α K, we deote by p α P irr the miimal polyomial of α over K. Give p P irr, we let Ω p K be the set of roots of p(x). Let be the absolute value v iduces o K. I geeral, give a ultrametric space (S, ), s S ad r R, r > 0, we let B(s, r) = {s S s s < r} be the ope ball of ceter s ad radius r. I this sectio we show that there is a oe-to-oe correspodece from W to the set of orbits of K uder the actio of the absolute Galois group G K = Gal( K/ K), that is, give α, α K, W α = W α if ad oly if α ad α are roots of the same irreducible polyomial over K. Equivaletly, the set W is i bijectio with the set P irr. Similarly, Ŵ is i bijectio with Pirr. We itroduce ow i these spaces suitable atural topologies. We edow W with the Zariski topology, that is, the topology which has as a ope basis the sets E(ϕ 1,..., ϕ ) = {W α ϕ i W α, i = 1,..., }, for ϕ i K(X), i = 1,...,, N (see [17, Chapt. VI, 17]). The set Ŵ is edowed with a similar topology: for ψ i K(X), i = 1,...,, we set Ê(ψ 1,..., ψ ) = {W α ψ i W α, i = 1,..., }. We edow the set P irr with the followig ultrametric distace, itroduced by Kraser (see [7]): for p, q P irr, we set (p, q) mi{ α β α Ω p, β Ω q } I other words, the fuctio (p, q) measures the smallest distace betwee the roots of p(x) ad the roots of q(x). As Kraser poits out, for each α Ω p there exists β Ω q such that α β = (p, q). The other mai result of this sectio is that with the topologies we have itroduced P irr, W ad Ŵ are homeomorphic. 10

11 We recall the followig formula due to Kraser (see [7, p ]): give p P irr of degree with set of roots Ω p = {α = α 1,..., α } K ad β K with miimal polyomial q P irr, we have p(β) = max{ (p, q), α α i } (3.1) i=1 Recall that, for ϕ K(X) ad p P irr, the valuatio of ϕ(α) i K, for α Ω p, does ot deped o the choice of α i Ω p. I fact, for α, α Ω p, α α, the elemets ϕ(α), ϕ(α ) are cojugated over K, i.e., ϕ(α ) = σ(ϕ(α)), for some σ G K; the sice K is complete of rak oe, v ad v σ coicide ([15, A., p. 127]), thus v(ϕ(α)) = v(σ(ϕ(α))) = v(ϕ(α )). The right had side of (3.1), which we deote by M p ( (p, q)), is a strictly icreasig fuctio of (p, q), that is, (p, q) < (p, q ) M p ( (p, q)) < M p ( (p, q )). I particular, formula (3.1) shows that p(β) depeds oly o p(x) ad (p, p β ). Therefore, p(β) = M p ( (p, p β )) = M p ( (p, q)) for each q P irr such that (p, p β ) = (p, q). More geerally, the real-valued fuctio M p (r) max{r, α α i } (3.2) i=1 is a strictly icreasig fuctio of the real variable r. It follows immediately that for each r R + we have M p (r) = p(β), for each β K such that (p, p β ) = r. Remark 3.1. We will use the followig well-kow fact: a ratioal fuctio ϕ K(X) is a cotiuous fuctio over K with respect to the v-adic topology o its domai of defiitio, that is, if α K is such that ϕ(α), the, for each ε R, ε > 0, there exists δ R, δ > 0, such that for all α B(α, δ) we have ϕ(α ) B(ϕ(α), ε). I particular, if ϕ is itegral at α (i.e., ϕ(α) V ) the for ε < 1, the correspodig δ is such that ϕ is itegral over B(α, δ), that is, ϕ W α, for all α B(α, δ). Actually, we ote that, sice we are cosiderig ratioal fuctios over K, if ϕ is itegral at α K ad p(x) = p α (X) P irr, the ϕ is itegral over the set Ω p, so, it is sufficiet that a elemet of Ω p is i B(α, δ) i order for ϕ to be itegral at α ; equivaletly, for all p α B(p α, δ) = {p P irr (p, p α ) < δ}, ϕ W α. We will also cosider the sets formed by the cotractio to K[X] ad to K[X] of the valuatio domais of W ad Ŵ, respectively: W K[X] = {W α K[X] α K}, Ŵ K[X] = {W α K[X] α K} These cotractios were first cosidered by MacLae i [11, p. 382], where they are called value rigs; see also [9] for a deeper study of their properties. Note also that we have the equality W α K[X] = {f K[X] f(α) V }, where the last rig is a rig of iteger-valued polyomials over the fiite set {α}, deoted by It K ({α}, V ) i [14]. The set W K[X] becomes a topological space whe it is edowed with the atural Zariski topology, where a basis is give by E K[X] (f 1,..., f ) = {W α K[X] f i W α K[X], i = 1,..., }, where f 1,..., f are elemets of K[X] ad N. Similar topology is give to Ŵ K[X]. We will show i our last mai theorem that also W K[X] ad Ŵ K[X] are homeomorphic to P irr, W ad Ŵ. We show first i the ext theorem that these sets are i bijectio with each other. 11

12 Theorem 3.2. Let α 1, α 2 K. The the followig coditios are equivalet: i) W α1 = W α2. ii) W α1 K[X] = W α2 K[X]. iii) W α1 K[X] = W α2 K[X]. iv) α 1, α 2 are cojugated over K. v) W α1 = W α2. I particular, the sets W, Ŵ, W K[X], Ŵ K[X] ad P irr are i bijective correspodece. Proof. i) ii). Clear. ii) iii). Suppose W α1 K[X] = W α2 K[X] ad let f W α1 K[X]. Let us write f(x) = h i=0 âix i, with â i K, 0 i h ad let us cosider a i K, 0 i h, such that v((â i a i )αj i ) = v(â i a i ) + iv(α j ) > v(f(α 1 )), for all 0 i h ad j = 1, 2. If g(x) = h i=0 a ix i K[X], the v(g(α 1 )) mi{v((g f)(α 1 )), v(f(α 1 ))} = v(f(α 1 )) 0 (ote that v((g f)(α 1 )) mi{v(â i a i )+iv(α 1 ) i {0,..., h}}). Thus, g W α1 K[X] = W α2 K[X] ad v(f(α 2 )) mi{v((f g)(α 2 )), v(g(α 2 ))} 0 (ote that v((f g)(α 2 )) mi{v(â i a i )+iv(α 2 ) i {0,..., h}} > v(f(α 1 ))). Hece, f W α2 K[X]. Sice f(x) was arbitrary, this shows that W α1 K[X] W α2 K[X] ad the other iclusio is proved i the same way. iii) iv). Suppose W α1 K[X] = W α2 K[X] ad let p = p α1 P irr. Let us fix P, 0. If p(α 2 ) 0, the there exists N such that p(α2) / V, which is a cotradictio, sice for every N we have p(x) W α1 K[X] = W α2 K[X]. Therefore p(α 2 ) = 0, so that α 1, α 2 are cojugated over K. iv) v). Suppose there exists σ G K such that σ(α 1 ) = α 2. Give f W α1, σ(f) = f W α2, so W α1 W α2. The other iclusio is proved symmetrically, so W α1 = W α2. v) i). Sice W αi K(X) = W αi, i = 1, 2, the claim follows immediately. The last statemet is ow clear. Propositio 3.3. Let α K, r > 0 ad P, 0. The there exist q K[X] ad N such that q(x) is itegral at α ad for all W α E( q(x) ), we have p α B(p α, r). I particular, the family {E( q(x) ) q K[X], N} is a subbasis for the Zariski topology o W. Proof. Give α K, let p = p α P irr ad d the degree of p(x). Let B = B(p, r), where r is ay give positive real umber. We suppose first that p(x) is separable. We choose N such that < M p (r). Let (p) be the miimum distace of the distict roots α = α 1,..., α d of p i K. The there exists δ > 0 such that if q K[X] is moic of degree d ad q p G < δ (where G is the absolute value associated with the Gauss valuatio v G ), the for each α i Ω p, i = 1,..., d, there exists a uique root β i Ω q such that α i β i < mi{ (p), r} (see [15, Chapt. 5, p. 139]). I particular, (q, p) < r ad q(x) is irreducible over K (hece also over K; this holds by Kraser s Lemma, see [15, Chapt. 5, G. p. 139]). Moreover, up to a choice of a smaller δ, we may also suppose that q(α), so that the polyomial q(x) is itegral at α. We claim ow 12

13 that M p (ρ) = M q (ρ) for each ρ R, ad by the very defiitio (3.2) it is sufficiet to show that α 1 α i = β 1 β i, for each i = 2,..., d. Ideed, we have β 1 β i = β 1 α 1 + α 1 α i + α i β i = α 1 α i sice α 1 α i (p) > β j α j for j = 1, i. Fially, we have ( ) q(x) W α E q(α ) = M q ( (q, p α )) = M p ( (q, p α )) < M p (r) (q, p α ) < r sice M p ( ) is a strictly icreasig fuctio. Therefore, (p α, p) max{ (p α, q), (q, p)} < r, thus, p α B. We suppose ow that p(x) is iseparable. Let l > 0 be the characteristic of K ad let p(x) = p(x lm ), where m 1 ad p P irr is separable. Note that if Ω p = {α = α 1,..., α t } (where t < d is the umber of distict roots of p(x)), the Ω p = {α lm 1,..., α lm t }. We set γ = α lm ad γ i = α lm i, for i = 2,..., t. Let r = r lm. By the first part of the proof, there exist q P irr K[X] ad N such that q(x) is itegral at γ = α lm ad for each W γ E( q(x) ) we have p γ B( p, r). We set q(x) = q(x lm ). We have that q(x) is itegral at α sice q(α) = q(γ). Moreover, let W α E( q(x) This implies that W γ E( q(x) ), where γ = α lm. Therefore, (p γ, p) < r. Now, (p γ, p) = mi{ γ γ i i = 1,..., t} = = mi{ α lm α lm i i = 1,..., t} = mi{ α α i lm i = 1,..., t} = (p α, p) lm Hece, (p α, p) < r, as wated. For the last statemet, the fiite itersectios of the sets E( q(x) ), for q K[X] ad N, form a basis for a topology o W which is weaker tha the Zariski topology o W. Let E(ϕ), ϕ K(X) be a elemet of the subbasis for the Zariski topology. Let W α E(ϕ), α K. By cotiuity of ϕ, there exists a eighborhood B = B(p α, r) of p α (X), r > 0, such that for all p α B, we have ϕ W α (see Remark 3.1). The by what we have proved above, there exists a basic ope set E = E( q(x) ), where q K[X] ad N, such that W α E E(ϕ), which shows that {E( q(x) ) q K[X], N} is a subbasis for the Zarisky topology o W. Note that i the case K is a separable extesio of K, the above proof shows that we may also suppose that the polyomials q(x) above are irreducible over K (hece also over K). Also, the q(x) same Propositio shows that {Ê( π ) q K[X], N} is a subbasis for the Zarisky topology o Ŵ. These results are the mai igrediets for the proof of our last theorem. Theorem 3.4. The topological spaces W, Ŵ, W K[X], Ŵ K[X] ad P irr are homeomorphic. Proof. Let I be the bijectio from Ŵ to W defied by W α W α, for each α K (Theorem 3.2). Clearly, this map is cotiuous, sice for ϕ K(X) we have I 1 (E(ϕ)) = {W α W α ϕ} = {W α W α ϕ} = Ê(ϕ). Coversely, let Ê(ψ), ψ K(X), be a basic ope set ad let W α be a elemet of I(Ê(ψ)). Let P, 0. By Propositio 3.3, there exist q K[X] ad N such that W α E( q(x) ) I(Ê(ψ)). Hece, I is ope ad W ad Ŵ are homeomorphic. ). 13

14 The same proof shows that the maps W W K[X], W α W α K[X], ad Ŵ Ŵ K[X], W α W α K[X], for α K, are homeomorphisms (they are bijectios by Theorem 3.2). Fially, we prove that Φ : P irr W, p α W α, α K, is a homeomorphism. We prove first that Φ is cotiuous. It is sufficiet to show that, for ay ϕ K(X), the preimage via Φ of E(ϕ) i P irr is ope. Let q P irr be a elemet of this preimage, so that ϕ W β, where β is ay root of q(x). We have to show that there exists a eighborhood of q(x) i P irr which is cotaied i Φ 1 (E(ϕ)), that is, if (p, q), p P irr, is sufficietly small, the Φ(p) = W β is i E(ϕ), where β K is ay root of p(x). But this holds because a ratioal fuctio ϕ is a cotiuous fuctio with respect to the v-adic topology o its domai of defiitio, see Remark 3.1. Next, we show that Φ is a ope map. Let B(p, r) be a ope eighborhood of a give p P irr for some r R, r > 0, ad let W α Φ(B(p, r)) (p α, p) < r. By well-kow properties of ultrametric distaces, B(p, r) = B(p α, r). Hece, without loss of geerality, we may suppose that p(α) = 0. Let P, 0. By Propositio 3.3, there exist q K[X] ad N such that W α E( q(x) ) Φ(B(p, r)), so the map Φ is ope, hece a homeomorphism. Ackowledgemets The author warmly thaks the referee for his/her mostly valuable suggestios which allow to state the mai results i greater geerality. The author wishes also to thak Bruce Olberdig for precious ad isightful commets which improved the quality of the paper. Refereces [1] V. Alexadru, N. Popescu, A. Zaharescu, All valuatios o K(X). J. Math. Kyoto Uiv. 30 (1990), o. 2, [2] N. Bourbaki. Algèbre commutative, Herma, Paris, [3] J.-L. Chabert, G. Perugielli, Polyomial overrigs of It(Z), J. Commut. Algebra 8 (2016), o. 1, [4] R. Gilmer. Multiplicative ideal theory, Corrected reprit of the 1972 editio, Quee s Uiversity, Kigsto, ON, [5] R. Gilmer, W. Heizer, D. Latz, W. Smith, The rig of iteger-valued polyomials of a Dedekid domai. Proc. Amer. Math. Soc. 108 (1990), o. 3, [6] I. Kaplasky, Maximal fields with valuatios. Duke Math. J. 9, (1942), [7] M. Kraser, Nombre des extesios d u degré doé d u corps p-adique, Les Tedaces Géométriques e Algèbre et Théorie des Nombres, Ed. CNRS, Paris, 1966, pp [8] F.-V. Kuhlma, Value groups, residue fields, ad bad places of ratioal fuctio fields. Tras. Amer. Math. Soc. 356 (2004), o. 11, [9] K. A. Loper, F. Tartaroe. A classificatio of the itegrally closed rigs of polyomials cotaiig Z[X]. J. Commut. Algebra 1 (2009), o. 1,

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