Definable Valuations on NIP Fields

Transcription

1 Definable Valuations on NIP Fields Dissertation submitted for the degree of Doctor of Natural Science Presented by Katharina Dupont at the Faculty of Science Department of Mathematics and Statistics Date of the oral examination: 2015/02/17 First referee: Salma Kuhlmann Second referee: Assaf Hasson Konstanzer Online-Publikations-System (KOPS) URL:

2

3

4

5 Contents Preface iii Deutsche Zusammenfassung (German Summary) ix 1 Definable Valuations The Valuation Ring O G Induced by a Subgroup G Criteria for the Non-Triviality of O G Criteria for the Definability of O G O G for G = (K ) q and G = K (p) O G on NIP fields Further Work A Appendix: Valuation Theory and Absolute Values B Appendix: Field Theory C Appendix: Real Fields D Appendix: Model Theory E Appendix: Measure Theory Notations Index Bibliography i

6

7 Preface Valuation theory has been studied since the beginning of the twentieth century and since then applications to many areas of mathematics have been found. A detailed history which covers approximately the years 1912 until 1940 has been written by Roquette (see [Roq]). We call a valuation ring O definable if there exists a formula ϕ in the language L ring := {+,, ;0,1} (possibly with parameters) such that O = {x K ϕ(x)}. A valuation is called definable if the corresponding valuation ring is definable. First results about definable valuations were already obtained by J. Robinson. From her proof that the integers are definable in the rational numbers follows that for any prime q the q-adic valuation on the rational numbers is definable (see Example 1.2). Further she showed that for any field K the valuation ring K[[t]] on K((t)) is existentially definable. Later Ax showed that this valuation ring is parameter-free definable. In [Koe94] Jochen Koenigsmann first shows that for any prime q, every not q-closed field with a non-trivial q-henselian valuation admits a non-trivial definable valuation, unless q = 2 and the field is euclidean. He further follows from this that if a field is not separably closed and not real closed and admits a t-henselian topology then it admits a definable valuation. Recent results, several by Koenigsmann himself and students of Koenigsmann, refine Koenigsmann s theorems. In [Koe04, Section 3.5] Koenigsmann gives conditions under which t-henselian valuations are definable without parameters. Anscombe and Koenigsmann show in [AK14] that for certain primes p the valuation ring F p [[t]] on F p ((t)) is parameter-free existentially definable. These results were generalized by Fehm in [Feh15] to certain henselian fields with finite, pseudo finite, pseudo real closed or pseudo algebraically closed residue field. In this paper also some positive as well as negative results on uniform definability can be found. Motivated by this work in [Pre] Prestel gives model theoretic criteria for uniform - and -definability of valuation rings on henselian fields. In[CDLM13] Cluckers, Derakhshan, Leenknegt and Macintyre give positive and negative results on uniform definability and parameter-free definability for henselian valued fields with finite and pseudo finite residue fields. In [JK15a] Jahnke and Koenigsmann give conditions on the residue field and the Galois group, under which the henselian valuation is definable without parameters. They as well give a similar improvement for q-henselian valuations. In her thesis [Jah13] Jahnke as iii

8 Preface well examines under which conditions a henselian field admits a parameter-free definable henselian valuation. In [Hon14] Hong generalizes a long-known result for discrete henselian valuations and gives an explicit parameter-free formula defining a certain q-henselian valuation. All these results have in common that they assume the existence of a non-trivial henselian or q-henselian valuation or a t-henselian topology. In [Kru15] Krupiński obtains a result with assumptions of a different nature. He shows that under the assumption that K is a field such that for every finite field extension L/K ( L : (L ) q) < for every prime q N and ( L : L (p)) < for p := char(k), where L (p) denotes the Artin-Schreier group, either K admits a non-trivial definable valuation or every non-trivial valuation on K has divisible value group and algebraically closed residue field. For fields of characteristic zero he obtains some partial results of this kind. A very active direction in recent model theoretic research are structural questions about fields under different model theoretic assumptions. Krupiński s work was motivated by understanding the structure of superosy fields. In [She09, Conjecture 5.34] Shelah makes a structural conjecture on strongly-dependent fields. He conjectures that stronglydependent fields are either real closed or algebraically closed or somehow p-adic like. This motivated Hasson to the following conjecture. 0.1 Conjecture. Let K be an infinite NIP field. Assume that ( L : (L ) q) < for every finite field extension L/K and every prime q N. Then K is algebraically closed or K is real closed or K admits a non-trivial definable valuation. In this thesis we suggest an approach to Conjecture 0.1 using V-topologies. A topology T is called a V-topology if for N the set of zero neighbourhoods of T the following axioms hold: (V1) N := U N U = {0} and {0} / N (V2) U, V N W N W U V (V3) U N V N V V U (V4) U N x, y K V N (x+v) (y +V) x y +U (V5) U N x K V N (x+v) 1 x 1 +U (V6) U N V N x, y K x y V x U y U We will be using the methods developed in[koe94]. In[Koe94] Koenigsmann defines on a field K a valuation ring O G induced by a subgroup G of K. He gives exact criteria under which there exists a formula ϕ such that O G is defined by ϕ for all (K,G ) (K,G). We will be following Koenigsmann s arguments to show that under certain assumptions there exists a definable valuation ring inducing the same topology as O G. The difference iv

9 will be in the tools used to show the non-triviality of O G and hence of the definable valuation. Despite the nice results and useful tools developed in [Koe94], the work is only available as a preprint with very brief proofs and some missing arguments. Furthermore when working with [Koe95] Jahnke found gaps in the proof of Proposition 1.4 (b) and the proof of Theorem 3.2. In [Koe94, Theorem 3.1] Koenigsmann in the so called weak case applies [Koe95, Proposition 1.4] and gives a very similar proof as for [Koe95, Theorem 3.2]. In personal discussion with Jahnke we found out that the proof of [Koe94, Theorem 3.1] is not completely correct either. Concerning the proof of [Koe95, Proposition 1.4 (b)] Chatzidakis has recently suggested how to fix the gap. The proof she suggests is quite complex and currently a student of her is writing it up in detail as part of his master thesis. As the complete proof is not available yet, we will avoid the use of Part (b) of the proposition. Applying results from [JK15b] we will give an alternative proof of [Koe94, Theorem 3.1] for the weak case in Chapter 5, which uses only Part (a) and (c) of [Koe95, Proposition 1.4] (see Theorem 5.19). Koenigsmann s work is of great importance for our work and the proofs given in [Koe94] are, as mentioned above, very brief. We will therefore not only include the alternative proof of [Koe94, Theorem 3.1] but also give detailed proofs for the results leading to the theorem, as well as for [Koe94, Theorem 4.1] (see Corollary 5.24). Mainly we will follow the ideas given in [Koe94], reorganizing the material to better suit our purpose. However some changes have been made in Chapter 3 which includes the proof of [Koe94, Theorem 2.5]. In his proof Koenigsmann refers to Svenonius Definability Theorem, a consequence of Beth s Theorem. In [Sve59] Svenonius gives conditions under which we get a disjunction of explicit definitions of a relational symbol. Koenigsmann does not explain how from this follows that O G is defined by the same formula for all (K,G ) (K,G). We will alternate the proof using Beth s Theorem instead of Svenonius Theorem. In this way we obtain a full proof of [Koe94, Theorem 2.5] (see Theorem 4.8). In Chapter 1 we will start with some preliminaries on definable valuations. InChapter 2 wewilldefineforeveryadditiveormultiplicativesubgroupofafieldk the valuation ring O G and investigate some of its properties. Here we are mostly following [Koe94, Section 1]. In Chapter 3 we will give criteria under which O G is a non-trivial valuation ring. In the first section we will define the topology T G induced by a subgroup G. We will show that for a proper subgroup G of K, T G it is a V-topology if and only if O G is non-trivial. This section includes Definition 2.2, Proposition 2.3, Corollary 2.4 and the second part of Proposition 1.4 of [Koe94]. As defined above T G is a V-topology if it fulfills the axioms (V1) to (V6). This is used in [Koe94] to show that the non-triviality of O G can be expressed by a first-order formula. We suggest that this result could also be used to show the non-triviality of O G under various assumptions. Hence in the second section we will investigate the axioms further for T G, when G a multiplicative subgroup of K. We will be able to show that for this topology the first part of Axiom (V1), Axiom (V2) and v

10 Preface Axiom (V 5) hold without further conditions and reduce the other axioms considerably. In Chapter 4 we will give exact criteria under which there exists an L G -formula ϕ whichdefineso G forall(k,g ) (K,G). L G heredenotesanextensionofthelanguage L ring := {+,, ;0,1} by a unary relation symbol G. This chapter includes the alternate proof of [Koe94, Theorem 2.5] mentioned above. In Chapter 5 we will investigate O G for the Artin-Schreier group G = K (p) and the group of q-th powers (K ) q for q char(k) prime. We will show that K admits a definable valuation inducing the same topology as O G, unless q = 2 and K is euclidean. In Section 5.1 we will define q-henselian valuations. We will show the existence of a non-trivial definable valuation for fields which are not q-closed and admit a non-trivial q-henselian valuation, again unless q = 2 and K is euclidean. In Section 5.2 we will define t-henselian topologies and show the existence of a non-trivial definable valuation on every non separably closed, non real closed field which admits a t-henselian topology. This chapter is based on [Koe94, Theorem 3.1] and [Koe94, Theorem 4.1] and includes the changes mentioned above. In Chapter 6 we will introduce NIP theories. In the second section we will show that Axiom (V1) holds for (K ) q K under the assumptions of Conjecture 0.1. With what we have shown before follows Corollary 6.36: Assume K fulfills the assumptions of Conjecture 0.1 and further 1 K and for some q char(k) prime G := (K ) q K and ζ q K. Let N G := { n i=1 a i (G+1) n N,a 1,...,a n K }. If the axioms (V3) V N G V V G+1 (V4) V N G V V G+1 (V6) V N G x,y K x y V x G+1 y G+1. hold, then K admits a non-trivial definable valuation. In the third section we will investigate finite field extensions of NIP fields further. We will show that a non algebraically closed and non real closed field fulfilling the assumptions of the conjecture, admits a finite field extension L such that 1 L and for some q char(l) prime (L ) q L and ζ q L. In the last section of the chapter we will investigate stable fields, a subclass of NIP fields. Chapter 7 gives an overview of the questions we are working on at the moment and projects we consider for the future. For the convenience of the reader we will finish with a rather elaborate appendix. The aim of the appendix is mainly to remind the reader of some of the theorems and definitions we are using. For most of the theorems we will give only references, however we will give some proofs that cannot be found in standard textbooks even for some well known and easy to prove facts. While in Chapters 2, 4 and 5 as well as in Section 3.1, we work with multiplicative subgroups as well as with additive subgroups, in Section 3.2 and Chapter 6 we concentrate only on multiplicative subgroups. It has been shown in [KSW11] that NIP fields vi

11 are Artin-Schreier closed, i.e. K (p) = {x p x x K} = K, where p = char(k). Therefore in the NIP context we cannot apply the results we obtain in Chapter 5 for the Artin-Schreier group but only the results for the group of q-th powers for q char(k) prime. As the main motivation in Section 3.2 Chapter 6 was to make progress towards the proof of Conjecture 0.1, in this parts of the thesis we concentrate on multiplicative subgroups and in particular groups of qth-powers. vii

12

13 Deutsche Zusammenfassung Wir untersuchen die Existenz nicht-trivialer definierbarer Bewertungen auf NIP Körper. Ein Bewertungsring O heißt definierbar, wenn es eine Formel ϕ in der Sprache L ring := {+,, ;0,1} gibt, so dass O = {x K ϕ(x)}. Eine Bewertung heißt definierbar, wenn der zugehörige Bewertungsring definierbar ist. Die Bedingung NIP ist eine kombinatorische Bedingung aus der Modelltheorie. Wir erläutern diese in Kapitel 6 ausführlich. Unsere Arbeit ist durch die folgende Vermutung motiviert: 1 Vermutung. Sei K ein unendlicher NIP Körper. Für alle endlichen Körpererweiterungen L/K und alle Primzahlen q N sei ( L : (L ) q) <. Dann gilt genau einer der folgenden drei Fälle. K ist algebraisch abgeschlossen oder K ist reell abgeschlossen oder es existiert eine nicht-triviale definierbare Bewertung auf K. Wir entwickeln einen Ansatz, der uns Bedingungen für die Existenz einer definierbaren nicht-trivialen Bewertung liefert, die für modelltheoretische Methoden und Voraussetzungen zugänglich sind. Hierfür wenden wir V-Topologien an. Eine Topologie T is eine V-Topologie wenn für die Menge der Nullumgebungen N von T die folgenden Axiome erfüllt sind (V1) N := U N U = {0} and {0} / N (V2) U, V N W N W U V (V3) U N V N V V U (V4) U N x, y K V N (x+v) (y +V) x y +U (V5) U N x K V N (x+v) 1 x 1 +U (V6) U N V N x, y K x y V x U y U. In [Koe94] definiert Koenigsmann zu jeder Untergruppe G auf einem Körper K einen Bewertungsring O G. Er zeigt dann unter welchen Kriterien eine Formel ϕ existiert, so dass ϕ für alle (K,G ) (K,G) den Bewertungsring O G definiert. Wir gehen wie in [Koe94] vor um die Existenz eines definierbaren Bewertungsrings zu zeigen, der die gleiche Topologie wie O G induziert. Unsere Arbeit unterscheidet sich von der von Koenigsmann hinsichtlich der Methoden, die benutzt werden um zu zeigen dass O G, und somit die definierbare Bewertung, nicht-trivial ist. ix

14 Deutsche Zusammenfassung (German Summary) Trotz der guten Resultate und nützlichen Werkzeuge, die in [Koe94] entwickelt werden, ist die Arbeit nur als Preprint mit sehr knappen Beweisen und einigen fehlenden Argumenten verfügbar. Außerdem wurde im Beweis von [Koe94, Satz 3.1] ein Fehler entdeckt und eine Lücke in einem der verwendeten Resultate aus[koe95] gefunden. Wir geben daher einen alternativen Beweis für einen Teil von [Koe94, Satz 3.1] an. Da Koenigsmanns Arbeit für unseren Ansatz von großer Bedeutung ist, werden wir auch die anderen Resultate ausführlich beweisen, wobei sich die Beweise in dieser Arbeit zum Teil von denen in [Koe94] unterscheiden. In Kapitel 1 geben wir eine Einführung zu definierbaren Bewertungen. In Kapitel 2 definieren wir zu jeder Untergruppe G eines Körpers den Bewertungsring O G und untersuchen einige seiner Eigenschaften. In Kapitel 3 untersuchen wir unter welchen Bedingungen O G nicht-trivial ist. Im ersten Abschnitt definieren wir die Topologie T G induziert von einer Untergruppe G. Wir werden zeigen, dass falls G eine echte Untergruppe ist, T G genau dann eine V-Topologie ist, wenn O G nicht trivial ist. T G is eine V-Topologie wenn die oben angegeben Axiome gelten. Zum einen folgt daraus, dass wir in der Logik erster Ordnung ausdrücken können, dass O G nicht-trivial ist. Zum anderen kann dieses Resultat auch direkt verwendet werden, um zu zeigen, dass O G nicht-trivial ist. Daher untersuchen wir die Axiome im zweiten Abschnitt für die Topologie T G genauer. Wir zeigen, dass für diese Topologie der erste Teil des Axioms (V1), so wie die Axiome (V2) und (V5) ohne weitere Voraussetzungen erfüllt sind. Die anderen Axiome vereinfachen wir deutlich. In Kapitel 4 zeigen wir wann eine L G -Formel ϕ, die für alle (K,G ) (K,G) den Bewertungsring O G definiert, existiert. L G ist hierbei eine Erweiterung der Sprache L ring := {+,, ;0,1} um ein einstelliges Relationssymbol G. In Kapitel 5 untersuchen wir O G für die Artin-Schreier Gruppe G = K (p) und die Gruppe der q-ten Potenzen G := (K ) q für q char(k) prim genauer. Außer für den Fall char(k) q = 2 und K euklidisch, zeigen wir, dass es auf K eine definierbare Bewertung gibt, die die selbe Topologie wir O G induziert. In Kapitel 6 definieren wir, wann ein Körper die Eigenschaft NIP hat. Im zweiten Abschnitt zeigen wir, dass das Axiom (V1) für (K ) q K unter den Bedingungen von Vermutung 1 erfüllt ist. Mit den Ergebnissen der vorherigen Kapitel bekommen wir damit das folgende Resultat: Angenommen K erfüllt die Bedingungen von Vermutung 1 und außerdem gilt 1 K und für eine Primzahl q char(k) ist G := (K ) q K und ζ q K. Sei N G := { n i=1 a i (G+1) n N,a 1,...,a n K }. Falls die Axiome (V3) V N G V V G+1 (V4) V N G V V G+1 (V6) V N G x,y K x y V x G+1 y G+1 erfüllt sind, so existiert eine nicht-triviale definierbare Bewertung auf K. x

15 Im dritten Abschnitt untersuchen wir endliche Körpererweiterungen von NIP Körpern. Wir zeigen, dass ein nicht algebraisch abgeschlossener und nicht reell abgeschlossener Körper, der die Voraussetzungen von Vermutung 1 erfüllt eine endliche Körpererweiterung L besitzt, so dass 1 L und für eine Primzahl q char(l) gilt (L ) q L und ζ q L. Wir schließen das Kapitel mit einem Abschnitt über stabile Körper, einer Teilklasse der NIP Körper, ab. Kapitel 7 gibt einen Überblick über die Fragen an denen wir zur Zeit arbeiten und einen Ausblick auf weitere Projektideen. xi

16

17 1. Definable Valuations The aim of this chapter is to give a short introduction to definable valuations, to prepare the ground for the following chapters. We will give a few examples of fields which do not admit definable valuations. Our aim is not to give a complete list. With the first two examples, algebraically closed and real closed fields, we will see that the three cases in Conjecture 0.1 are indeed distinct. Example 1.5 follows from Corollary 6.59, but as the proofs of Example 1.5 and Example 1.7 are very concrete we want to include them in this section nevertheless. There are many other interesting aspects we will not or only very briefly mention here, including parameter-free definability and complexity of the defining formula. The results in this chapter are all known, although for some of them it is hard to find the proof in full detail and, in particular for discrete henselian valuations, in the generality given here. The proofs given here shall help the reader to get a good understanding of definable valuations. We are interested in valuations definable in the language of rings L ring := {+,, ;0,1} (possibly with parameters). However as it is sometimes useful to consider extensions of this language, the following definition is more general. In particular in Chapter 4 we will consider the language L G := {+,, ;0,1;G}, where G is a unary relation symbol. In the following by L(K) we will denote the extension of the language L by a constant for every element of K. 1.1 Definition. Let L be a language. Let K be a field. Let O be a valuation ring on K. (a) We call O L-definable (with parameters) or definable in L, if there exists an L(K)-formula ϕ(x) such that O = {x K ϕ(x)}. We say ϕ defines O. (b) We call a valuation v on a field K L-definable if O v is L-definable. (c) We say O (respectively v) is L-definable without parameters or parameter-free L-definable if ϕ as above is an L-formula. (d) We call O (respectively v) definable if O is L ring -definable. 1.2 Example. For every prime number q N the q-adic valuation is definable in Q. Proof: In [Rob49, Theorem 3.1] J. Robinson shows that there exists an L ring -formula ϕ such that for every x Q we have ϕ(x) if and only if x Z. Further O vq = { a b a,b Z,b 0,q b}. 1

18 1. Definable Valuations ( Let ψ(x) := a,b ϕ(a) ϕ(b) x b = a b = 0 ( n(ϕ(n) q n = b) )). Then O vq = {x Q ψ(x)}. In the following lemma we will give a class of examples for which we can write down a formula defining a non-trivial valuation explicitly. We will show a more general result on the existence of a definable valuation in Chapter 5, i.e. if a non real closed and non separably closed field admits a non-trivial henselian valuation it admits a non-trivial definable valuation. The formula from Part (a) of the following lemma is often cited for (Q q,v q ). The formula given in Part (b) can be found in [Ax65]. We will give detailed proofs here, in order to get a better understanding of the example. 1.3 Lemma. Let v : K Γ { } be a discrete henselian valuation. Let 1 Γ be the minimal positive element and let c K with v(c) = 1. (a) v is definable with parameters. We have O v = { x K y y 2 y = c x 2}. (b) If Γ is archimedean then v is definable without parameters. O v = { ( x K d yy 2 y = d x 2 z y ( y 2 y = d z 2) ( a, b yy 2 y = d a 2 b 2 y ( y 2 y = d a 2) y ( y 2 y = d b 2)))}. Proof: (a) Let us first assume x, y K are such that y 2 y = c x 2. Suppose x / O v. Suppose v(y) < 0. Then v ( y 2 y ) = min { v ( y 2), v(y) } = 2 v(y) is even. But v ( c x 2) = v(c)+v ( x 2) = 1+2 v(x) is odd. This contradicts y 2 y = c x 2. Hence v(y) 0. Therefore v ( y 2 y ) min { v ( y 2), v(y) } 0. As v(x) < 0 we have v(x) 1. Hence v ( c x 2) = v(c)+2 v(x) = 1+2 v(x) 1 2 = 1 < 0. This again contradicts y 2 y = c x 2. Therefore if there exists y K such that y 2 y = c x 2 then x O v. On the other hand if x O v we have f := X 2 X c x 2 O v [X] with f = X 2 X and f = 2 X 1 and therefore 0 is a simple root of f. By Hensel s Lemma (Theorem A.47 (v)) f has a simple root y O v and for this y y 2 y = c x 2 holds. (b) For every d K define O d := { x K yy 2 y = d x 2}. Let A := {O d O d K and O d is closed under multiplication}. 2

19 We claim that O v = A = {O d O d K and O d is closed under multiplication} = {x K d(x O d O d K O d is closed under multiplication)} { ( = x K d yy 2 y = d x 2 z y ( y 2 y = d z 2) ( a, b yy 2 y = d a 2 b 2 y ( y 2 y = d a 2) y ( y 2 y = d b 2)))}. By (a) O v = O c A, as O c is a non-trivial valuation ring. Therefore O v A. We will now show O v A. Let d K with O d closed under multiplication and O d O v. We will show that O d = K. Let x O d \ O v. Then v(x) < 0. Let y K. As Γ is archimedean there exists m N such that 2 m v(x) < v(d)+2 v(y) and therefore (d (y v x m) ) 2 = v(d)+2 v ( y x m) = v(d)+2 v(y) 2 m v(x) > 0. Let f = X 2 X d (y x m) 2. With Hensel s Lemma (Theorem A.47 (v)) follows that f has a root in K, i.e. z z 2 z = d (y x m ) 2. By definition of O d follows that y x m O d. As x O d and O d is closed under multiplication y = x m y x m O d. Therefore O d = K. Altogether follows O v = { ( x K d yy 2 y = d x 2 z y ( y 2 y = d z 2) ( a, b yy 2 y = d a 2 b 2 y ( y 2 y = d a 2) y ( y 2 y = d b 2)))}. 1.4 Example. For any prime number q the q-adic valuation on Q q is parameter-free definable. As q = } {{} in this case already the formula given in (a) is parameter-free. q-times More generally if (K, v) is q-adically closed then v is a definable valuation on K (see [EP05, Page 157] for the definition). 3

20 1. Definable Valuations 1.5 Example. Let K be a field. If K is algebraically closed the only definable valuation is the trivial. Proof: As algebraically closed fields allow quantifier elimination in the language L ring (see [PD11, Theorem 3.3.4]) every valuation which is definable by an L ring (K)-formula is definable by a quantifier free L ring (K)-formula. Terms in L ring (K) in one variable are polynomials therefore atomic formulas in L ring (K) are of the form f 1 = f 2 for polynomials f 1,f 2 K[X] where without loss of generality we can assume f 2 = 0. As non-constant polynomials can only have finitely many roots, sets defined by formulas of this type are finite, empty or K. If we take the complements of finite sets we get cofinite sets. The intersection or union of finite and cofinite sets are again finite or cofinite sets. Therefore if ϕ and ψ define a finite or cofinite set, so do ϕ, ϕ ψ and ϕ ψ. Therefore every set which is defined by a quantifier free L ring (K)-formula is finite or cofinite. No non-trivial valuation ring is finite or cofinite and therefore valuation rings in algebraically closed fields cannot be definable. 1.6 Lemma. Let K be a field. If there exists a non-trivial definable valuation on K then there exists a non-trivial definable valuation on every structure which is elementary equivalent to K. If the valuation on K is definable without parameters then the valuations on the elementary equivalent structures are definable by the same parameter-free formula. Proof: A formula ϕ defines a non-trivial valuation ring on a field if and only if the following sentences hold NT (ϕ) : x ( ϕ(x) ) R 1 (ϕ): x,y ( ϕ(x) ϕ(y) ϕ(x y) ) R 2 (ϕ): x,y ( ϕ(x) ϕ(y) ϕ(x+y) ) R 3 (ϕ): x ( ϕ(x) ϕ( x) ) R 4 (ϕ): ϕ(1) ( B(ϕ): x ϕ(x) y ( x y = 1 ϕ(y) )). The formula NT expresses that the set defined by ϕ is not the whole field, R 1 to R 4 say that the defined set is a ring and B that it is a valuation ring. Let K K. Suppose ϕ is an L ring (K)-formula such that O = {x K ϕ(x)} is a non-trivial valuation ring on K. Letϕ bethel ring -formulawhichweobtainbyreplacingtheparametersofϕbyvariables z 1,...,z n. 4

21 Then z 1,...,z n NT (ϕ ) R 1 (ϕ ) R 2 (ϕ ) R 3 (ϕ ) R 4 (ϕ ) B(ϕ )isanl ring -sentence which holds in K. As K and K are elementary equivalent it holds as well in K. Assume we have NT (ϕ ) R 1 (ϕ ) R 2 (ϕ ) R 3 (ϕ ) R 4 (ϕ ) B(ϕ )(c 1,...,c n ) for c 1,...,c n K. Let ϕ be the L ring (K )-formula which we obtain by replacing z 1,...,z n by the parameters c 1,...,c n in ϕ. Then O = {x K ϕ (x)} is a non-trivial definable valuation on K. If ϕ is parameter-free then ϕ = ϕ = ϕ. Now we can show with a similar proof as for algebraically closed fields that there is no non-trivial definable valuation on real closed fields. We will prove this first for archimedean ordered real closed fields. The general case follows from the last theorem and the completeness of the theory of real closed field. 1.7 Example. On a real closed field only the trivial valuation is definable. Proof: Let (K, ) be an archimedean ordered real closed field. As real closed fields allow quantifier elimination in the language L = {+,, ;0,1; } (see [PD11, Theorem 4.2.3]), every valuation which is definable by an L ring (K)-formula is definable by a quantifier free L (K)-formula. As in the proof of Example 6.10 we can assume without loss of generality that atomic formulas are either of the form f(x,y) = 0 or f(x,y) 0 for a polynomial f K[X,Y]. Both kinds of formulas define sets which are finite unions of intervals. If we take the complement of a finite union of intervals or the intersection or union of two finite unions of intervals we get again a finite union of intervals. Therefore if ϕ and ψ define a finite union of intervals so do ϕ, ϕ ψ and ϕ ψ. Therefore every set which is defined by a quantifier free L (K)-formula is a finite union of intervals. Let O be a valuation ring on K with O = 1 i n a i,b i for some a i K { } and b i K { } where denotes either ( or [ and denotes ) or ], i.e. a i,b i can be an open, half-open or closed interval. As O is a ring we have N O and therefore there exits 1 j n with a j,b j N infinite and therefore b j =. Suppose there exists x K with x / O. As (K, ) is archimedean there exists m N with m+x > a j and therefore m+x a j, ) = a j,b j O. But this is a contradiction as v(m+x) = min{v(x),v(m)} v(m) 0 = v(x) < 0. Therefore O is trivial. The case of arbitrary real closed fields now follows from Lemma 1.6 and the fact that the theory of real closed fields is complete (see [PD11, Theorem 4.2.2]), i.e. all models of the theory are elementary equivalent and hence every real closed field is elementarily equivalent to the archimedean real closed field R. 1.8 Theorem. Let L/K be a finite field extension. If O is a non-trivial definable valuation ring on L then O K is a non-trivial definable valuation ring on K. 5

22 1. Definable Valuations Proof: From Lemma A.1 follows that if O is non-trivial, then O K is also non-trivial. Let α 1,...,α n L be a K-basis of L. For every β L there exist unique a 0,...,a n with β = a 1 α a n α n. We therefore get a bijective map from L to the n-vector space K... K by sending β to (a 1,...,a n ). We now want to define addition and multiplication on K n. We define (a 1,...,a n )+(b 1,...,b n ) := (a 1 +b 1,...,a n +b n ). We further define (a 1,...,a n ) := ( a 1,..., a n ). If β 1 = a 1 α 1 + +a n α n and β 2 = b 1 α 1 + +b n α n then β 1 β 2 = a 1 b 1 α 2 1 +(a 1 b 2 +a 2 b 1 )α 1 α 2 + +a n b n α 2 n = n i,j=1 a i b j α i α j. As α i α j L for all 1 i,j n there exist c (k) ij α i α j = c (1) ij α 1 + +c (n) ij α n. We get β 1 β 2 = ( n i,j=1 a n ) i b j k=1 c(k) ij α k Thereforewedefine(a 1,...,a n ) (b 1,...,b n ) = This gives us what we want. K (1 k n) with ( n ) i,j= 1 a i b j c (k) ij α k. = n k=1 ( n i,j=1 a i b j c (1) ij,..., n i,j=1 a i b j c (n) ij We will now identify each term in L ring (L) with an n-tuple of terms in L ring (K). The free variable x we will identify with (x 1,0,...,0). (This makes sure that we get elements of K.) Every other variable y we will identify with (y 1,...,y n ). (Without loss of generality we assume that x only occurs as free variable.) 0 we will identify with (0,...,0). 1 we will identify with (1,0,...,0). For c = c 1 α c n α n L we will identify the parameter c with ( c 1,...,c n ). If t and s are terms which we identify with (t ( 1,...,t n ) and (s 1,...,s n ) we can identify n ) t+s with (t 1 +s 1,...,t n +s n ) and t s with i,j=1 t i s j c (1) ij,..., n i,j=1 t i s j c (n) ij and t with ( t 1,..., t n ) as above. Now we will assign a formula ψ to every formula ϕ recursively. If t and s are terms which we identify with (t 1,...,t n ) and (s 1,...,s n ) we assign to the formula t = s the formula t 1 = s 1... t n = s n. Let ϕ 1 and ϕ 2 be formulas, ψ 1 and ψ 2 are the assigned formulas and y identified with (y 1,...,y n ) is a variable. To ϕ 1 ϕ 2 we assign ψ 1 ψ 2, to ϕ 1 we assign ψ 1 and to y ϕ 1 we assign y 1,..., y n ψ 1. Assume ϕ defines a non-trivial valuation O on L. Let ψ be the formula assigned to ϕ as described above. Then it is easy to see that ψ defines O K. ). 6

23 1.9 Remark. Let L/K be a finite field extension. Let O be a parameter-free definable valuation on L. Let α 1,...,α n L be a K-basis of L. For i,j {1,...,n} let c (1) ij,...,c(n) ij K such that α i α j = c (1) ij α 1 + +c (n) ij α n. If c (k) ij Z for all i,j,k {1,...,n} then O K is parameter-free definable. InparticularifL = K( 1)orL = K(ζ q )forsomeprimeq, theno K isparameter-free definable. Proof: From the proof of Theorem 1.8 follows that the parameters needed to define O K are c (1) ij,...,c(n) ij K such that α i α j = c (1) ij α 0+ +c (n) ij α n for i,j {1,...,n} and the parameters needed to define O. Therefore O K is parameter-free definable if O is parameter-free definable and c (k) ij (i,j,k {1,...,n}) are constant L ring -terms. But the constant L ring -terms are exactly the elements from Z. 7

24

25 2. The Valuation Ring O G Induced by a Subgroup G The aim of this chapter is to introduce the valuation ring O G induced by an additive or multiplicative subgroup G of a field K. For this we will define when a valuation is compatible, weakly compatible and coarsely compatible with a subgroup G. O G will be defined as the intersection of all valuation rings which are coarsely compatible with G. We will show that with this definition O G is a valuation ring which is coarsely compatible with G. Further we will define a case distinction for O G that will reappear throughout the thesis. For this case distinction we will show some general facts. Some of the following definitions and theorems will be slightly different for additive and multiplicative subgroups. Often instead of repeating everything we will write the differences for multiplicative subgroups in square brackets [...] if there is no danger of misunderstanding. If we say G is a subgroup of K, this can mean either a subgroup of the additive group (K,+) or the multiplicative group (K, ), unless explicitly otherwise noted. We will say G is a proper subgroup of K if G K [G K ]. 2.1 Definition. Let K be a field. Let O be a valuation ring on K. Let G be a subgroup of K. Let M denote the maximal ideal of O. (a) O is compatible with G if and only if M G [1+M G]. (b) O is weakly compatible with G if and only if there exists an O-ideal A with A = M such that A G[1+A G]. (c) O is coarsely compatible with G if and only if v is weakly compatible with G and there is no proper coarsening Õ of O such that Õ G. Let v be a valuation on K. We call v compatible (respectively weakly compatible, coarsely compatible) with G if and only if O v is compatible (respectively weakly compatible, coarsely compatible) with G. The following easy to prove facts will be used throughout the thesis. 2.2 Facts. Let O be a valuation ring on a field K and G a subgroup of K. (a) Let O be compatible with G and Õ O be a coarsening of O. Then Õ is also compatible with G. (b) Let O be compatible with G. Then O is weakly compatible with G. 9

26 2. The Valuation Ring O G Induced by a Subgroup G (c) If O is weakly compatible with G and Õ O is a proper coarsening of O then Õ is compatible with G. (d) Let G K be an additive subgroup of K and O G. Then O G. (e) Let O G. Then O is compatible with G. (f) Let O be weakly compatible with G. Then there exists a coarsening of O which is coarsely compatible with G. (g) Let O be weakly compatible with G. If O G then O is coarsely compatible with G. (h) Let O be weakly compatible with G and O G. Then O is coarsely compatible with G if and only if O is maximal with O G. Proof: Let M denote the maximal ideal of O. (a) Let M denote the maximal ideal of Õ. Then M M G and hence Õ is compatible with G. [ 1+ M ] 1+M G (b) Let A := M. Then A fulfills the conditions in Definition 2.1 (b) and hence O is weakly compatible with G. (c) As O is weakly compatible with G there exists an O-ideal A such that A = M and A G [1+A G]. Let M denote the maximal ideal of Õ. By Lemma A.38 M A and therefore M [ A G 1+ M ] 1+A G. Hence Õ is compatible with G. (d) Let x O\O = M. We have 1+x O G. As 1 O G and G is closed under addition it follows that x = 1+x+( 1) G. This shows O \O G and hence, as O G by assumption, O G. (e) If G is an additive subgroup of K, from O G follows by (d) O G and therefore M O G. If G is a multiplicative subgroup of K, we have 1+M O G. Hence in both cases O is compatible with G. (f) If O is already coarsely compatible with G there is nothing to show. If O is a valuation ring which is weakly compatible but not coarsely compatible with G, there exists a proper coarsening O of O such that (O ) G. A maximal such valuation ring is a coarsening of O which is coarsely compatible with G. (g) Assume O is not coarsely compatible with G. Then for some proper coarsening of Õ we have Õ G. As O Õ we have O G. But this contradicts the assumption. (h) This follows directly from the definition. 10

27 2.3 Lemma. Let (K, v) be a valued field with residue characteristic q. Let G be a subgroup of K. Let v be weakly compatible with G. Then there exists n N such that q n M v G [1+q n M v G]. Proof: As v is weakly compatible with G there exists an O v -ideal A with A G [1+A G] and A = M v. As char ( K ) = q we have q M v and therefore there exists n N such that q n A. Let x q n M v. Then v(x) > v(q n ) and therefore by Claim A.30 (a) x A. Hence q n M v G [1+q n M v 1+A G]. We will define the valuation ring O G induced by a subgroup G as the intersection over all coarsely compatible valuation rings. To show that O G is a valuation ring, we will show that the valuation rings which are coarsely compatible with G are linearly ordered. For the proof we will need a lemma which follows from the Approximation Theorem (see Theorem A.24). In order to give a detailed proof we will first show the following: 2.4 Lemma. Let O 1, O 2 be independent valuation rings on a field K. Let A 1 be a non-trivial O 1 -ideal and A 2 a non-trivial O 2 -ideal. Then K = A 1 +A 2 and K = (1+A 1 ) (1+A 2 ). Proof: We will first show that (b 1 +c 1 A 1 ) (b 2 +c 2 A 2 ) for all b 1,b 2 K and c 1,c 2 K. Let a 1 A 1 \ {0} and a 2 A 2 \ {0}. For i = 1, 2 let v i : K Γ i { } be a valuation corresponding to O i. From the Approximation Theorem (Theorem A.24) follows that there exists x K such that v i (x b i ) > v i (a i c i ) = v i (a i ) + v i (c i ) for i = 1,2. Thus v i ( x bi x b i c i c i ) = v i (x b i ) v i (c i ) > v i (a i ). By Claim A.30 follows A i and hence x b i + c i A i. Therefore x (b 1 +c 1 A 1 ) (b 2 +c 2 A 2 ) and hence (b 1 +c 1 A 1 ) (b 2 +c 2 A 2 ). To show K = A 1 +A 2 let x K. Then (x A 1 ) A 2, i.e. there exist a 1 A 1 and a 2 A 2 such that x a 1 = a 2. Thus x = a 1 +a 2 A 1 +A 2. Therefore K = A 1 +A 2. Now to show K = (1+A 1 ) (1+A 2 ) let x K. Then (x+x A 1 ) (1+A 2 ). Hence there exist a 1 A 1 and a 2 A 2 such that x (1+a 1 ) = x + x a 1 = 1 + a 2. We have x = (1+a 1 ) 1 (1+a 2 ) (1+A 1 ) 1 (1+A 2 ). By Lemma A.40 we have (1+A 1 ) 1 = 1+A 1 and hence x (1+A 1 ) (1+A 2 ). As 0 / (1+A 1 ) (1+A 2 ) we have K = (1+A 1 ) (1+A 2 ). 11

28 2. The Valuation Ring O G Induced by a Subgroup G Applying this lemma on the residue field of the finest common coarsening of two valuations, we obtain the following: 2.5 Lemma. Let O 1 and O 2 be two non-comparable valuation rings on a field K. Let O be the finest common coarsening of O 1 and O 2 and M the maximal ideal of O. Let A 1 be an O 1 -ideal with M A 1 and A 2 an O 2 -ideal with M A 2. Then O = A 1 +A 2 and O = (1+A 1 ) (1+A 2 ). Proof: Let K := O/M be the residue field of O and : O K the residue homomorphism. Then (O 1 ) and (O 2 ) are independent valuation rings on K and (A 1 ) and (A 2 ) are non-trivial ideals of (O 1 ) and (O 2 ) by Fact A.19 (c). By Lemma 2.4 we get K = (A 1 ) + (A 2 ) = (A 1 +A 2 ). For every x O we have (x) K = (A 1 +A 2 ) and therefore x A 1 +A 2 +M A 1 +A 2 +A 1 A 2 A 1 +A 2, as M A 1 A 2 by assumption. Hence x A 1 +A 2. Therefore O A 1 +A 2. As O is closed under addition we have A 1 +A 2 O and hence A 1 +A 2 = O. Again with Lemma 2.4 (b) we get K = ( 1+ (A 1 ) ) (1+ (A 2 ) ). For every x O we get (x) K = ( 1 + (A 1 ) ) (1 + (A 2 ) ) = ((1+A 1 ) (1+A 2 ) ) and hence x (1+A 1 ) (1+A 2 )+M. As 0 M M 1 M+A 1 M+A 2 M+A 1 A 2 M = (1+A 1 ) (1+A 2 ) M. (2.1) Therefore x (1+A 1 ) (1+A 2 )+M (2.1) (1+A 1 ) (1+A 2 )+(1+A 1 ) (1+A 2 ) M (1+A 1 ) (1+A 2 ) (1+M) A.40 (1+A 1 ) (1+A 2 ), where in the last step we are using that 1 + A 2 is a multiplicative group with 1+M 1+A 2 by Lemma A.40. Hence O (1+A 1 ) (1+A 2 ). Again by Lemma A.40 we have 1+A 1, 1+A 2 O and as O is closed under multiplication (1+A 1 ) (1+A 2 ) O. Altogether we get (1+A 1 ) (1+A 2 ) = O. 2.6 Lemma. Let G be a subgroup of a field K. Then any two valuation rings which are coarsely compatible with G are comparable. 12

29 Proof: Let O 1 and O 2 be two valuation rings on K which are weakly compatible with G. For i = 1,2 let M i be the maximal ideal of O i. Further let A i be an O i -ideal with A i G [1+A i G] and A i = M i. Suppose O 1 and O 2 are not comparable. Let O be the finest common coarsening of O 1 and O 2. By assumption O 1 O and O 2 O. Let M be the maximal idel of O. From Lemma A.38 follows that M A 1 and M A 2 and therefore by Lemma 2.5 we have O = A 1 +A 2 and O = (1+A 1 ) (1+A 2 ). Hence O O = A 1 +A 2 G+G = G [O = (1+A 1 ) (1+A 2 ) G G = G]. Therefore O is a proper coarsening of O 1 and O 2 with O G and hence by definition O 1 and O 2 are not coarsely compatible with G. We will now define the valuation ring O G. 2.7 Definition and Theorem. Let G be a subgroup of a field K. Let O G := {O O coarsely compatible with G}. Then O G is a valuation ring on K. We call O G the valuation ring induced by G. By M G we denote the maximal ideal of O G. Proof: Let A := {O O coarsely compatible with G}. Let x,y O G. Then for all O A we have x,y O and therefore x+y,x y O. Hence x+y,x y O G. Let x K such that x / O G. Then there exists O A such that x / O. As O is a valuation ring x 1 O. Let O A. By Lemma 2.6 O and O are comparable. If O O we have x / O and therefore as O is a valuation ring we havex 1 O. For O O we have x 1 O O. Thus x 1 O for all O A and hence x 1 O G. 2.8 Lemma. Let G be a subgroup of a field. (a) We have M G = { M M is the maximal ideal of a valuation ring (b) O G is coarsely compatible with G. which is coarsely compatible with G }. Proof: Let as above A := {O O coarsely compatible with G} and B := {M M is the maximal ideal of a valuation ring O A}. (a) Let x M G. Then x 1 / O G and hence x 1 / O for some O A. Let M be the maximal ideal of O. Then x M and M B. Hence x B. On the other hand let x B. Then there exists M B such that x M. Let O be the valuation ring with maximal ideal M. Then x 1 / O and O A. Hence x 1 / A = O G. Therefore x M G. 13

30 2. The Valuation Ring O G Induced by a Subgroup G Hence M G = B = { M M is the maximal ideal of a valuation ring which is coarsely compatible with G }. (b) We will first show that O G is weakly compatible with G. For every O A let M O be the maximal ideal of O and let A O be an O-ideal with AO = M O and A O G [1+A O G]. Define A G := {A O O A}. We show that A G is an O G -ideal with A G = M G and A G G [A G G]. Let a,b A G and x O G. There exist O 1,O 2 A such that a A O1 =: A 1 and b A O2 =: A 2. By Lemma 2.6 O 1 and O 2 are comparable. Without loss of generality let O 1 O 2. If O 1 O 2 by Lemma A.38 A 2 M 2 A 1. If O 1 = O 2 then A 1 = A 2. In both cases A 2 A 1 and therefore a,b A 1. As A 1 is an ideal a+b A 1 A G. Further x O G and therefore x O for every O A, especially x O 1. Therefore x a A 1 A G. Hence A G is an O G -ideal. We now show A G = M G. For every valuation O A by (a) A O M O M G. Hence A G M G. As M G is prime, A G M G. On the other hand let x M G. From (a) follows that there exists O A such that x M O = A O. Therefore there exists an n N such that x n A O A G and hence x A G. As A O G [1+A O G] for every O coarsely compatible with G we have A G G [1+A G G]. Hence O G is weakly compatible with G. Assume O G is not coarsely compatible with G. Then by definition there exists a valuation ring O such that O G O and O G. Without loss of generality let O be maximal with this property. Then O is coarsely compatible with G. Let x O \ O G. By definition of O G there exists a valuation ring Õ A with x / Õ. By Lemma 2.6 Õ and O are comparable. Therefore as x O \Õ we have Õ O. But this is a contradiction as then Õ cannot be coarsely compatible with G by definition. This shows that O G is coarsely compatible with G. 2.9 Remark. By definition O G is the finest valuation ring which is coarsely compatible with G. 14

31 In Lemma 2.6 we have shown that any two valuation rings which are coarsely compatible with the same proper subgroup are comparable. We will now show in Lemma 2.11 a slightly weaker result for valuation rings which are weakly compatible with the same proper subgroup. While in this case the valuation rings are not necessarily comparable, they are always dependent, i.e. there exists a non-trivial common coarsening. A valuation ring O is weakly compatible but not coarsely compatible with a group G if and only if there exists a proper coarsening Õ of O such that Õ G. Therefore in the following Lemma we will investigate valuation rings O 1 and O 2 in Part (a) for the case O 1,O 2 G and in Part (b) for the case O 1 G, O 2 G and O 2 weakly compatible with G Lemma. Let K be a field. Let G be a subgroup of K. Let O 1 and O 2 be valuation rings on K. (a) Let O 1,O 2 G. Then there exists a common coarsening O of O 1 and O 2 with O G [O G]. (b) Let O 1 G, O 2 G and let O 2 be weakly compatible with G. Then O 1 O 2. Proof: (a) If O 1 and O 2 are comparable either O 1 or O 2 is a common coarsening of O 1 and O 2. If O 1 and O 2 are not comparable let O be the finest common coarsening. Let M 1 and M 2 be the maximal ideals of O 1 and O 2. As O 1 and O 2 are not comparable, for the maximal ideal M of O we have M M 1 and M M 2. Hence by Lemma 2.5 O = M 1 +M 2 G [ O (1+M 1 ) (1+M 2 ) O 1 O 2 G]. (b) By Fact 2.2 (g) O 2 is coarsely compatible with G. Let O be a maximal valuation ring with O 1 O and O G. By Fact 2.2 (g) O is compatible with G. It follows directly from the definition, that O is coarsely compatible with G and therefore by Lemma 2.6 comparable with O 2. As O G and O 2 G we have O 2 O. Hence O 1 O O Lemma. Let G be a proper subgroup of a field K. Let O 1 and O 2 be two non-trivial valuation rings which are weakly compatible with G. Then O 1 and O 2 are dependent. Proof: If O 1 and O 2 are coarsely compatible with G then by Lemma 2.6 O 1 and O 2 are comparable and hence dependent. If O 1 is not coarsely compatible with G then by Fact 2.2 (g) O 1 G. If O 2 G then by Lemma 2.10 (a) there exists a common coarsening O of O 1 and O 2 with O G [O G]. As by assumption G K [G K ] O is non-trivial and hence O 1 and O 2 are dependent. If O 2 G then by Lemma 2.10 (b) we have O 1 O 2 and hence O 1 and O 2 are dependent. Analogous we show that O 1 and O 2 are dependent if O 2 is not coarsely compatible with G. 15

32 2. The Valuation Ring O G Induced by a Subgroup G It is often useful to distinguish three cases. We will use the names introduced in [Koe94], i.e. groupcase,weakcaseandresiduecase. InLemma2.13,Lemma2.14andLemma2.15 we will show some facts for each of this cases. In Corollary 2.17 this will be summarized Remark and Notation. Let O be a valuation ring on a field K and G a subgroup of K. Let : K O/M =: K denote the residue homomorphism. Then G := (G) is a subgroup of the residue field K. For the group case we get the following lemma: 2.13 Lemma. Let G be a subgroup of a field K. Assume that there exists a valuation ring O on K with O G. Then (a) O G G. (b) O G is the only valuation ring with this property which is coarsely compatible with G. (c) The valuation rings O with O G are exactly the refinements of O G. (d) All valuations which are weakly compatible with G are compatible with G. [ (e) G = K G = K ]. (f) If G is multiplicative, it is determined by its values, i.e. G = v 1( v(g) ). Proof: (a) Let Õ := {O O valuation ring O G}. We will first show that Õ is a valuation ring with Õ G. Then we will show that O G = Õ. Let x,y Õ. Then there exist O 1, O 2 {O O G} such that x O 1 and y O 2. By Fact 2.10 (a) there exists a common coarsening O of O 1 and O 2 with O G. As x,y O we have x+y,x y O and therefore x+y,x y Õ. Further if x Õ then x O for some valuation ring O such that O G. As O is a ring we have x O Õ. Hence Õ is a ring. By assumption there exists a valuation ring O such that O G and therefore O Õ. Hence Õ is a coarsening of a valuation ring and therefore as well a valuation ring. Now let x Õ. Then x, x 1 Õ and hence there exist O 1,O 2 with O 1,O 2 G such that x O 1 and x 1 O 2. By Fact 2.2 (a) there exists a common coarsening O of O 1 and O 2 with O G and hence x,x 1 O. We therefore have x O G. Hence Õ G. We now want to show O G = Õ. By Fact 2.2 (e) Õ is compatible with G and therefore weakly compatible with G. By the definition of Õ there does not exist O Õ such that O G. Therefore Õ is coarsely compatible with G. As O G is the intersection of all valuation rings which are coarsely compatible with G we have O G Õ. But as Õ G and O G is by Lemma 2.8 coarsely compatible with G from this already follows O G = Õ. Hence O G = Õ G. 16

33 (b) By Lemma 2.6 all valuation rings which are coarsely compatible with G are comparable. Therefore it follows directly from the definition that there can be at most one valuation ring O with O G which is coarsely compatible with G. (c) In the proof of part (a) we have shown O G = {O O G} hence the valuation rings O with O G are exactly the refinements of O G. (d) Let O be weakly compatible with G. If O G by Fact 2.2 (e) O is compatible with G. If O G by (a) and Lemma 2.10 (b) we have O G O and hence by Fact 2.2 (c) O is compatible with G. (e) Let a K = (O G /M G ). Then a = x for some x O G \ M G = O G G and [ hence x G. In the additive case 0 G and therefore 0 G. Therefore G = K G = K ]. (f) We have to show v 1( v(g) ) G. Let x v 1 (v(g)). Then there exits) γ v(g) with v(x) = γ. As γ v(g) v(y) = γ for some y G. We have v( x y = v(x) v(y) = 0 and thus x y O G G. From y G and x y G follows x = y x y G. Hence G = v 1( v(g) ). The second case will be called weak case Lemma. Let G be a subgroup of a field K. Assume there exists a valuation ring on K which is weakly compatible but not compatible with G. (a) There is no valuation ring O with O G. (b) O G is weakly compatible but not compatible with G and no other valuation ring is weakly compatible but not compatible with G. Proof: (a) This follows at once from Lemma 2.13 (d). (b) Let O be weakly compatible but not compatible with G. We will show that O = O G. As by (a) O G by Fact 2.2 (g) O is coarsely compatible with G. From the definition of O G follows O G O. By Fact 2.2 (c) follows O G = O as otherwise O would be compatible with G. By assumption there exists a valuation ring which is weakly compatible but not compatible with G. Hence O G is weakly compatible but not compatible with G and O G is the only valuation ring with this property. 17