Klein approximation and Hilbertian fields

J. reine angew. Math. 676 (2013), 213—225

DOI 10.1515/crelle.2012.007

Journal fur die reine undangewandte Mathematik( Walter de Gruyter

Berlin � Boston 2013

Klein approximation and Hilbertian fields

By Arno Fehm at Konstanz and Elad Paran at Ra’anana

Abstract. The quotient field of a generalized Krull domain of dimension exceedingone is Hilbertian by a theorem of Weissauer. Building on work of R. Klein we generalizethis criterion for Hilbertianity to a wider class of domains. This allows us to extend recentresults on Hilbertianity of fields of power series and obtain new Hilbertian fields.

1. Introduction

Recall that a field K is Hilbertian if for every irreducible polynomial f ðt;XÞ A K½t;X �,separable in X , there exists t A K such that f ðt;XÞ A K ½X � is irreducible, cf. [La], Chap-ter VIII, [FJ], Chapter 12. This notion stems from Hilbert’s Irreducibility Theorem, assert-ing that number fields have this property. The motivation for Hilbert was the inverseGalois problem—if a field K is Hilbertian, then for a finite group G to be realizable as aGalois group over K it su‰ces for it to be realizable over the field of rational functionsKðX Þ. Numerous Galois theoretic results, over di¤erent types of fields, were obtained viathis approach. It is thus important to determine which fields are Hilbertian.

In [Wei], Weissauer used methods of logic to prove a remarkably general theorem:The quotient field of a generalized Krull domain of dimension exceeding 1 is Hilbertian.A domain D is called a generalized Krull domain if there exists a family F of real valuationson the quotient field K ¼ QuotðDÞ that satisfies certain independence properties (cf. Nota-tion 3.1 below). Generalized Krull domains were first studied by Ribenboim [Ri1] in apurely algebraic context. Intuitively, these are domains having ‘‘good arithmetic’’, definedby the valuations in F, which play the role that prime elements play in unique factorizationdomains.

In recent years, Weissauer’s theorem served as a crucial ingredient in the proofs ofseveral Galois theoretic results over quotient fields of complete domains (for example in[HS], [Pa2], [Po2]). The prototype for such a domain is the ring DJXK of formal powerseries over a domain D. If D is not a field, then DJXK has dimension exceeding 1, and if Dsatisfies certain conditions (like being a Krull domain, cf. Section 3), then DJXK is a gener-alized Krull domain. However, in general DJXK need not be a generalized Krull domain,even if D is (see [PT], Theorem 1.1). This posed a di‰culty in obtaining Galois theoreticresults over F ¼ QuotðDJXKÞ, for certain domains D. In [FJ], Problem 15.5.9 (b), Jardenasks whether F is Hilbertian whenever D is a generalized Krull domain.

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The work [Pa4] gives a positive answer to this question. The proof of [Pa4],Theorem 2.4, applies Weissauer’s theorem indirectly, by constructing an overring R ofDJXK which is a generalized Krull domain of dimension exceeding 1, thus deducing thatQuotðRÞ ¼ QuotðDJXKÞ is Hilbertian. However, since the quotient field is determined bythe ring, one may ask what are the arithmetic properties of the ring DJXK itself that implythe Hilbertianity of its quotient field. Moreover, the construction used in [Pa4] reliesheavily on the fact that DJXK is a complete domain, a property that has no counterpart inWeissauer’s theorem, which requires only ‘‘arithmetic’’ properties (in particular, themethod of [Pa4] cannot handle rings lying between D½X � and DJXK).

In this work we identify the arithmetic property of the ring DJXK that yields theHilbertianity of its quotient field—we introduce the technical notion of a Krull ideal

(Definition 3.4) and observe the following:

(1) If R is a domain and p is a prime element of R that satisfiesTif1

Rpi ¼ f0g (e.g. theelement p ¼ X in DJXK), then Rp is a Krull ideal.

(2) In a generalized Krull domain every non-zero prime ideal is Krull.

We then prove the following generalization of Weissauer’s theorem:

Main Theorem. If an integrally closed domain has a non-maximal Krull ideal, then its

quotient field is Hilbertian.

Intuitively, Weissauer’s theorem asserts that a domain (of dimension exceeding 1)that has ‘‘globally good arithmetic’’, reflected by the fact that all of its prime ideals areKrull, has a Hilbertian quotient field. The Main Theorem asserts that it su‰ces that thering has ‘‘some good arithmetic’’, i.e. has at least one (non-maximal) Krull ideal. Forexample, by (1) above we get the following result, which was previously unknown:

Corollary A. Let R be an integrally closed domain containing a prime element p, suchthat

Tif1

hpi i ¼ f0g. If hpi is not maximal, then QuotðRÞ is Hilbertian.

From Corollary A we deduce the following generalization of the results of [Pa4]:

Corollary B. Let D be a domain, contained in a rank-1 valuation ring of its quotient

field, and let R be a domain satisfying

D½X �LRLDJXK:

Then QuotðRÞ is Hilbertian.

This removes the ‘‘unexpected’’ completeness condition which [Pa4] relies on. Furthergeneralizations are given in Section 4. Our proof of the Main Theorem relies on a paper ofR. Klein [Kl], who proved an approximation theorem for non-Hilbertian valued fields, andapplied his results to reprove Weissauer’s theorem.

214 Fehm and Paran, Klein approximation and Hilbertian fields



2. Klein’s approximation result for non-Hilbertian fields

We start with an explanation of Klein’s terminology and present his main result,which is an approximation result for non-Hilbertian fields—similar, and in some aspectseven stronger than the well-known strong approximation theorem for global fields.

Klein starts with the following characterization of Hilbertian fields:

Lemma 2.1 ([Kl], Folgerung 1). A field K is Hilbertian if and only if for every poly-

nomial f A K ½t;X � that satisfies

(*) f is monic and separable in X , the pole of t is unramified in the splitting field of f

over KðtÞ, and the irreducible factors of f in K ½t;X � are absolutely irreducible and non-linear

in X ,

there exists t A K such that f ðt;XÞ has no zero in K.

Therefore, if a field K is non-Hilbertian, then one can fix a polynomial f A K½t;X �satisfying (*) such that f ðt;XÞ has a zero in K for all t A K. If K is the quotient field ofa domain R, then (as Klein observes in [Kl], proof of Satz 3) one can assume in additionthat f has coe‰cients in R:

Lemma 2.2. If R is a domain and K ¼ QuotðRÞ is non-Hilbertian, then there exists a

polynomial f A R½t;X � that satisfies (*) and f ðt;X Þ has a zero in K for all t A K.

By an absolute value on a field K we mean as usual a norm (archimedean ornon-archimedean) which is multiplicative. In order to state his approximation result,Klein makes the following definition. Let P be the prime ring of K, for f A K ½t;X �denote by bf LK the set of coe‰cients of f , and if j � j is an absolute value on K letj f j :¼ maxfjbj : b A bf g.

Notation 2.3 ([Kl], Definition 3). For a polynomial f A K ½t;X �, numbers v;w A N

and d > 0, and an element t A K let

Kðv;d;wÞf ðtÞ

be the set of all h A K that satisfy the following properties:

(i) h is integral over P½bf ; t� of degreee v.

(ii) jhje dj f jw for every absolute value j � j on K with jtje 1.

For a finite set j � j1; . . . ; j � js of absolute values on K we can view K (and its subset

Kðv;d;wÞf ðtÞ) as diagonally embedded into the product space

Qss¼1

ðK; j � jsÞ, where each

ðK; j � jsÞ is just K equipped with the topology defined by j � js. Klein’s main result can thenbe formulated as follows:

215Fehm and Paran, Klein approximation and Hilbertian fields



Proposition 2.4 ([Kl], Satz 4). Let K be a non-Hilbertian field, let f A K ½t;X � satisfy(*) such that f ðt;XÞ has a zero in K for all t A K, and let s A N. Then there exist numbers

v;w A N and d > 0 such that for every s absolute values j � j1; . . . ; j � js on K and every element

t A K with jtjs > 1, 1e se s, the following holds:

1

tis a limit point of K

ðv;d;wÞf ðtÞ in

Qss¼1

ðK; j � jsÞ:

Since we are not so much interested in absolute values but in prime ideals and valu-ations, we will work just with a special case of Proposition 2.4. By a real valuation on K wemean a (Krull) valuation v on K whose value group vðK�Þ is a subgroup of R. (As usual,we denote by R� the group of units of a ring R.) Note that the valuation rings of realvaluations are precisely the rank-1 valuation rings. If we fix a > 1, then jxj :¼ a�vðxÞ definesa non-archimedean absolute value on K, and this yields a one-to-one correspondencebetween non-archimedean absolute values on K (modulo equivalence) and real valuationson K (modulo equivalence). The condition jtj > 1 then translates to vðtÞ < 0.

Proposition 2.5. Let R be a domain with a non-Hilbertian quotient field K , and let S

be a finite set of real valuations on K. Let t A K with vðtÞ < 0 for each v A S, and let A be the

integral closure of R½t�. Then 1

tcan be approximated (with respect to the set S) by elements

from A.

Proof. Let f A R½t;X � be a polynomial as in Lemma 2.2. By Proposition 2.4 we may

approximate1

t(with respect to S) by elements from the set K

ðv;D;wÞf ðtÞ, for a suitable choice

of v, D and w. Each element in that set is by definition integral over P½bf ; t�LR½t�, hencebelongs to A. r

As a first illustration of the strength of this result we give a simple criterion forHilbertianity that immediately implies the well-known fact that function fields areHilbertian:

Corollary 2.6. Let R be an integrally closed domain and w a non-trivial real valuation

on F ¼ QuotðRÞ such that wðxÞe 0 for each x A R. Then F is Hilbertian.

Proof. Since w is non-trivial on F , it is non-trivial on R, hence there exists t A R with

wðtÞ < 0. If F is non-Hilbertian, then Proposition 2.5 gives x A R with w x� 1

t

� �> w

1

t

� �,

hence wðxÞ ¼ �wðtÞ > 0, contradicting the assumption. r

Example 2.7. If K is a field, then F ¼ KðXÞ is Hilbertian.

Proof. Let R ¼ K ½X � and let w be the degree valuation on F given by vðXÞ ¼ �1.By Corollary 2.6, F is Hilbertian. r

Although the rest of this work deals with Hilbertianity of quotient fields of do-mains exceeding one, this example shows that Proposition 2.5 can also be applied toone-dimensional domains.




3. A generalization of Weissauer’s theorem

In this section we use Klein’s approximation result to prove our Main Theorem.

Notation 3.1. If R is an integral domain, then by an integral valuation on R we shallmean a non-trivial real valuation on the quotient field of R, whose valuation ring con-tains R. If v is an integral valuation on R, we shall denote its valuation ring in F ¼ QuotðRÞby Rv, and we denote by

pðvÞ :¼ fx A R : vðxÞ > 0g

the center of v on R. Following [Ri1], we say that v is essential on R if Rv equals the local-ization RpðvÞ.

Also following [Ri1], if F is a family of real valuations on a field F , and if Rv denotesthe valuation ring of v in F for each v A F, then we say that the domain R ¼

Tv AF

Rv is

defined by F (in particular, each v A F is integral on R). We say that F is of finite type if

Fx :¼ fv A F : vðxÞ3 0g

is finite for each x A F�.

A domain R is called a generalized Krull domain if it is defined by a family F of finitetype of integral valuations on R, where each v A F is essential on R, [Gr]. The family F isuniquely determined by R (up to equivalence of valuations), [Ri1], Proposition 1, and iscalled the essential family of R. If all the valuations in F are discrete, then R is called aKrull domain.

The following lemmas give some basic properties of generalized Krull domains:

Lemma 3.2. Let R be a generalized Krull domain with essential family F.

(a) The family of minimal non-zero prime ideals of R coincides with the family

fpðvÞ : v A Fg of centers on R of valuations in F.

(b) Each non-zero prime ideal p of R contains a minimal non-zero prime ideal.

Proof. For (a) see [Ri1], Proposition 2, for (b) see [Ri1], p. 221. r

Lemma 3.3. Let p be a non-zero prime ideal of a generalized Krull domain R with

essential family F, and let a A Rnp. Then there exists b A p with Fa XFb ¼ j.

Proof. By Lemma 3.2, there exists a valuation w A F such that pðwÞL p. ThenfpðvÞgv AFa

W fpðwÞg is a finite family of distinct minimal non-zero (and hence incompa-rable) prime ideals of R. Thus pðwÞƒ

Sv AFa

pðvÞ, cf. [AM], Proposition 1.11 (i), so there

exists b A pðwÞn� S

v AFa

pðvÞ�

. In particular, b A pðwÞL p, and if v A Fa, then b B pðvÞ

implies that v B Fb, and therefore FaXFb ¼ j. r




Definition 3.4. Let p be a prime ideal of a domain R. We say that p is a Krull ideal

if for each x A Rnp there exists an overring D of R that satisfies the following conditions:

(1) D ¼T

v AFRv for a finite set F of integral valuations on R.

(2) R1

p

� �XD ¼ R for some 03 p A p.

(3) x A D�.

Example 3.5. If R is a domain and p a prime element of R satisfyingTyi¼1

hpii ¼ f0g,

then hpi is a Krull ideal. Indeed, for each x A Rnhpi, let D be the valuation ring Rhpi ofthe p-adic valuation on QuotðRÞ (defined on R by vpðaÞ ¼ supðk : pk j aÞ).

Another general example of Krull ideals is given by the following proposition:

Proposition 3.6. Let R be a generalized Krull domain. Then each non-zero prime ideal

of R is a Krull ideal.

Proof. Let F be the essential family of R, let p be a non-zero prime ideal of R, andlet x A Rnp. By Lemma 3.3, there exists p A p with FxXFp ¼ j. Let D ¼

Tv AFp

Rv. SincevðxÞ ¼ 0 for each v A Fp, x A D�. Moreover,

R1

p

� �XDL

� Tv AFnFp

Rv

�XD ¼

Tv AF

Rv ¼ R: r

We are now ready to prove our Main Theorem:

Theorem 3.7. Let R be an integrally closed domain, and let p be a Krull ideal of R.

If p is not a maximal ideal of R, then F ¼ QuotðRÞ is Hilbertian.

Proof. Suppose F is not Hilbertian. We show that p is maximal. Let x A Rnp, and letD, p be as in Definition 3.4. Write D ¼

Tv AF

Rv for a finite family F of integral valuations

on R. Since x A D�, vðxÞ ¼ 0 for each v A F.

Put F 0 ¼ fv A F : vðpÞ > 0g and D 0 ¼T

v AF 0Rv. Note that D 0 also satisfies

R1

p

� �XD 0 ¼ R. Indeed, suppose y A R

1

p

� �XD 0, and let v A FnF 0. Then vðpÞ ¼ 0, hence

vðyÞf 0. Thus y A D 0 XT

v AFnF 0Rv ¼

Tv AF

Rv ¼ D, hence y A R1

p

� �XD ¼ R. Replace F

with F 0 and D with D 0 to assume without loss of generality that vðpÞ > 0 for each v A F.

Put t ¼ x

p. Then vðtÞ < 0 for each v A F. By Proposition 2.5,

1

tcan be approximated

(with respect to F) by elements from the integral closure A of R½t�. Note that R½t�LR1

p

� �.

Since R is integrally closed, so is the localization R1

p

� �, thus A is contained in R

1

p

� �.




Take y A ALR1

p

� �, such that v y� 1

t

� �> vðpÞ for each v A F. Put z ¼ y

pA R

1

p

� �.

Then v z� 1

x

� �¼ v y� 1

t

� �� vðpÞ > 0 for each v A F. Since v

1

x

� �¼ �vðxÞ ¼ 0 we have

vðzÞ ¼ 0 for each v A F. Thus, z A R1

p

� �X

Tv AF

Rv ¼ R1

p

� �XD ¼ R.

Since vðxÞ ¼ 0 and v z� 1

x

� �> 0 for each v A F, also vðzx� 1Þ > 0. Thus there

exists n A N such that n � vðzx� 1Þ > vðpÞ for each v A F, hence

ðzx� 1Þn

pA R

1

p

� �X

Tv AF

Rv ¼ R:

Therefore, ðzx� 1Þn A hpiL p, and since p is prime, this implies that zx� 1 A p.

It follows that every x A Rnp is invertible modulo p, so R=p is a field. Consequently, pis a maximal ideal of R. r

Note that since every prime ideal containing a Krull ideal is obviously Krull itself, theMain Theorem of the Introduction is just a reformulation of this.

From Theorem 3.7 we immediately get three general criteria for Hilbertianity. Thefirst one is Weissauer’s theorem, the other ones are new criteria that do not follow fromWeissauer’s theorem.

Corollary 3.8 (Weissauer’s theorem). Let R be a generalized Krull domain of dimen-

sion exceeding 1. Then F ¼ QuotðRÞ is Hilbertian.

Proof. Let p be any non-zero prime ideal of R which is not maximal. By Prop-osition 3.6, p is a Krull ideal, so Theorem 3.7 implies that F is Hilbertian. r

Corollary 3.9. Let R be an integrally closed domain, and let p be a prime element of R

withTyi¼1

hpii ¼ f0g. If hpi is not a maximal ideal, then F ¼ QuotðRÞ is Hilbertian.

Proof. By Example 3.5, hpi is a Krull ideal, so the claim immediately follows fromTheorem 3.7. r

Corollary 3.10. Let R be an integrally closed domain and let w be an integral

valuation on R. Suppose 03 p A pðwÞ satisfies R1

p

� �XRw ¼ R. If pðwÞ is not a maximal

ideal of R, then F ¼ QuotðRÞ is Hilbertian.

Proof. Clearly, pðwÞ is a Krull ideal (take D ¼ Rw). Thus the claim follows fromTheorem 3.7. r




4. Hilbertianity of fields of power series

A simple example for a use of Corollary 3.9 where Weissauer’s theorem (Corol-lary 3.8) does not apply is the following:

Proposition 4.1. Let D be an integral domain which is not a field. If R ¼ DJXK is

integrally closed, then the quotient field of R is Hilbertian.

Proof. The element X is prime in R, andTyi¼1

hX ii ¼ f0g. Since R=hXi ¼ D is not a

field, hXi is not a maximal ideal of R. Thus the claim follows by Corollary 3.9. r

Remark 4.2. Let D be a generalized Krull domain which is not a field. If D iseven a Krull domain, then R ¼ DJXK is also a Krull domain, [Ma], Theorem 12.4 (iii),of dimension exceeding 1. However, if D is not a Krull domain, then R is not even ageneralized Krull domain, [PT], Theorem 1.1. Thus one cannot apply Weissauer’s theo-rem to R in order to prove that F ¼ QuotðRÞ is Hilbertian. In [FJ], Problem 15.5.9 (b),Jarden asks whether such a field F is nevertheless Hilbertian. In [Pa4], §2, this problemis answered positively, by constructing a two-dimensional generalized Krull domain R 0

with RLR 0LF , and then applying Weissauer’s theorem to R 0 in order to deduce thatF ¼ QuotðRÞ ¼ QuotðR 0Þ is Hilbertian. The construction of R 0 is rather complicated, andrelies on the theory of convergent power series over complete valued fields (in particular,the Weierstrass division and preparation theorems), as well as on technical results con-cerning the characterization of generalized Krull domains. Using Proposition 4.1, we geta simple proof of this result. Indeed, since a generalized Krull domain is an intersectionof rank-1 valuation rings, D is a completely integrally closed domain and hence so isR ¼ DJXK, [Bou], §5.1.4, Proposition 14. In particular, R is integrally closed. Thus viaCorollary 3.9 we obtained a short solution of the open problem of Jarden that [Pa4] workshard to resolve.

The above mentioned proof of [Pa4] yields a more general result than asked by [FJ],Problem 15.5.9 (b). Namely, for QuotðDJXKÞ to be Hilbertian, [Pa4], Theorem 2.4, assumesonly that D is a domain contained in a rank-1 valuation ring (i.e. the valuation ring of areal valuation) of its quotient field; in particular, D need not be integrally closed, which isnecessarily the case in Proposition 4.1. We now exploit Corollary 3.9 to reprove this resultas well, and extend it further to subrings of DJXK:

Proposition 4.3. Let D be a domain contained in a rank-1 valuation ring Dv of

K ¼ QuotðDÞ, and let R be a ring satisfying

D½X �LRLDvJXK:


Proof. Let F ¼ QuotðRÞ, and put R 0 ¼ DvJXKXF . Since DvJXK is integrallyclosed (see above), so is R 0. Since X is prime in DvJXK, X is also prime in R 0. Similarly,Ti

R 0X i ¼ f0g. Since DLR 0=R 0X LDv, R 0X is not a maximal ideal of R 0. Thus by

Corollary 3.9, F ¼ QuotðRÞ ¼ QuotðR 0Þ is Hilbertian. r




From this we immediately get the following result for power series in severalvariables:

Corollary 4.4. Let D be an integral domain, and let n > 1. Then the quotient field of

DJX1; . . . ;XnK is Hilbertian.

Proof. Let D0 ¼ DJX1; . . . ;Xn�2K. The intersection of the discrete valuation ringQuotðD0ÞJXn�1K with QuotðD0JXn�1KÞ contains DJX1; . . . ;Xn�1K. Therefore, the quotientfield of DJX1; . . . ;XnK ¼ DJX1; . . . ;Xn�1KJXnK is Hilbertian by Proposition 4.3. r

Using Corollary 3.10, we now further generalize Proposition 4.3, in two di¤erentdirections: We replace power series by generalized power series, and we remove the condi-tion that X is contained in R.



DLRLDvJXK and RƒDv:


Proof. Let F ¼ QuotðRÞ and pick f A RnDv. Since f ð0Þ A Dv and Dv LF ,f0ðX Þ :¼ f ðX Þ � f ð0Þ A ðF XDvJXKÞnDv. Write f0ðXÞ ¼ aX n þ X nþ1gðX Þ for some03 a A Dv, n > 0 and g A DvJXK.

Since a A DvLK , the map hðXÞ 7! hðaXÞ is a K-automorphism s of KððXÞÞ.Let F 0 ¼ sðFÞ. Then s

�f0ðX Þ

�¼ anþ1X nuðX Þ A F 0, where uðX Þ ¼ 1 þ XgðaX Þ A DvJXK�.

Since 03 a ¼ sðaÞ A Dv LF 0, the element pðXÞ :¼ X nuðXÞ belongs to S :¼ F 0XDvJXK.Since RLDvJXK we have sðRÞLDvJaXKLDvJXK, hence sðRÞLSLF 0. SinceF 0 ¼ s

�QuotðRÞ

�¼ Quot

�sðRÞ

�, it follows that QuotðSÞ ¼ F 0.

Since DvJXK is integrally closed (see above), so is S. Let w be the X -adic valuationon KððXÞÞ with valuation ring KJXK, and let Ow be the valuation ring of w in F 0. SinceuðXÞ A DvJXK�, we get that

S1

p

� �XOw ¼ F 0 XS

1

p

� �XOw LF 0 XDvJXK

1

p

� �XOw

¼ F 0 XDvJXK1

X n

� �XOwLF 0XDvJXK

1

X n

� �XKJXK

¼ F 0 XDvJXK ¼ S;

hence S1

p

� �XOw ¼ S. Since wðpÞ ¼ n > 0, p belongs to the center p of w on S. Since

Dv LSLDvJXK, S=p ¼ Dv, hence p is a non-maximal prime ideal of S. By Corollary 3.10,F 0 ¼ QuotðSÞ is Hilbertian, hence so is F ¼ s�1ðF 0Þ. r




Notation 4.6. If R is a ring and G an ordered cancellative commutative monoid, wedenote by RJX GK the ring of generalized power series f ðXÞ ¼

Pg AG

agXg with coe‰cients

ag A R and well-ordered support

suppð f Þ ¼ fg A G : ag3 0g:

For example, if we denote by Zf0 and Rf0 the non-negative integers resp. real numbers(with addition and the natural order), then RJX Zf0K ¼ RJXK is just the usual ring of formalpower series, while RJX Rf0K is a much larger ring that contains for example the ring of

formal Puiseux seriesSyn¼1

RJX1nK. If R is a field and G is an ordered abelian group, then

RJX GK is a field and carries a natural Henselian valuation v with value group G given byvð f Þ ¼ min

�suppð f Þ

�, [Ef], Theorem 2.8.2, Corollary 18.4.2.

Fields of generalized power series play an important role already in Krull’s seminalpaper [Kr]. Kaplansky then proved in [Ka] that they serve as universal domains of valua-tion theory. The systematic study of rings of generalized power series started with [Ri2].



D½X �LRLDvJXRf0K:


Proof. Let F ¼ QuotðRÞ, R 0 ¼ DvJXRf0KXF , let w be the natural valuation on

KJX RK, and let Ow ¼ KJX Rf0K be its valuation ring. By [Ri3], (5.3), DvJXRf0K is integrally

closed, hence so is R 0. Clearly, DvJXRf0K

1

X

� �XOw ¼ DvJX

Rf0K, hence

R 0 1

X

� �XOw ¼ F XR 0 1

X

� �XOw LF XDvJX

Rf0K1

X

� �XOw ¼ R 0;

so R 0 1

X

� �XOw ¼ R 0. The center pðwÞ of w on R 0 is not maximal, since DLR 0=pðwÞLDv.

Thus, Corollary 3.10 implies that F ¼ QuotðR 0Þ is Hilbertian. r

We want to conclude this section with a brief discussion for which rings D it holdsthat QuotðDJXKÞ is Hilbertian. By the above results this certainly holds whenever D is con-tained in a rank-1 valuation ring of K ¼ QuotðDÞ. The following observation shows that intypical situations this is generally the case:

Fact 4.8. If D is a domain such that K ¼ QuotðDÞ3D has finite transcendence

degree over its prime field F or over some field F contained in D, then D is contained in a

rank-1 valuation ring of K.

Indeed, by the Chevalley extension theorem [EP], Theorem 3.1.1, D is contained insome valuation ring of K , which necessarily has rank at most tr:degðK=FÞ þ 1 < y, [EP],Corollary 3.4.4, and thus has a coarsening of rank 1.




On the other hand, if D ¼ K is a field, then QuotðDJXKÞ ¼ KððXÞÞ is Henselian withrespect to the X -adic valuation and therefore not Hilbertian, [FJ], Lemma 15.5.4. If v is avaluation we denote by cfðvÞ the cofinality of the value group of v, that is, the cardinality ofa minimal cofinal subset, cf. [EP], p. 50. The following observation shows that it is not truethat QuotðDJXKÞ is Hilbertian whenever D is not a field.

Fact 4.9. Let v be a valuation on a field K with cfðvÞ > @0, and let D denote its

valuation ring. Then QuotðDJXKÞ ¼ KððX ÞÞ.

This can be proved just like the special case presented in [Pa3], Example 2.16. Thisfact also immediately answers negatively the following problem by Jarden: In [Ja], Prob-lem 3.8, he asked whether the inverse Galois problem holds for QuotðDJXKÞ whenever D isa domain of characteristic zero which is not a field. However, if D is a valuation ring ofrank 2@0 with quotient field C, then all finite extensions of QuotðDJXKÞ ¼ CððXÞÞ are cyclicby the Puiseux Theorem.

On the other hand, the following fact shows that the equality QuotðDJXKÞ ¼ KððXÞÞis rather exceptional, cf. [Pa1], Example 2.3 (c):

Fact 4.10. Let D be an integral domain that is contained in a valuation ring Dv of

K ¼ QuotðDÞ with cfðvÞ ¼ @0. Then QuotðDJXKÞkKððXÞÞ.

This leaves open the question whether QuotðDJXKÞ is Hilbertian in this case, when-ever D is not contained in a rank-1 valuation ring of K. The prototypical situation wherethis happens, and which seems of particular interest to us, is the following:

Question 4.11. Let D be a valuation ring of rank @0. Is QuotðDJXKÞ Hilbertian?

5. Rings of convergent power series

In this section we apply the results of the previous section in order to extend a Galoistheoretic result of Harbater.

Notation 5.1. Let A be a subring of Q and denote by AfXg the ring of those formalpower series in AJXK that converge on the complex open unit disc, cf. [Ha1], p. 803.

These and similar rings of convergent arithmetic power series play an important rolein Galois theory. For example, Harbater proved that the inverse Galois problem has a pos-itive answer for QuotðZfXgÞ: Every finite group occurs as the Galois group of a Galoisextension of QuotðZfXgÞ, [Ha2], Corollary 3.8.

Combining our results on Hilbertianity of fields of power series with known theoremsfrom Galois theory we can easily extend this result. We make use of the result of Pop thatthe inverse Galois problem has a positive answer for every field F for which the followingtwo conditions hold, cf. [Po1], [DD], §1:

(1) F is Hilbertian.

(2) F contains a so-called ample (or large, cf. [Po1], p. 2) field.




We now verify (1) and (2) for the fields QuotðAfXgÞ:

Proposition 5.2. For every subring A of Q that properly contains Z, the field

QuotðAfXgÞ contains an ample field.

Proof. Let R 0 be the ring of those f A AfXg that converge to a holomorphic func-tion on the open unit disc which extends to a continuous function on the closed unit disc.As in [FP], Lemma 3.1, one sees that R 0 is complete with respect to the norm k � k definedthere (that lemma states completeness of a family of rings ArJtK, 0 < r < 1, but the proofalso works for A1JtK ¼ R 0). If A3Z we can pick a A A with 0 < jaj < 1. Then the se-quence ðanX nÞn AN of elements of R 0 converges to zero with respect to k � k, hence [FP],Theorem 2.10, implies that QuotðR 0ÞLQuotðAfXgÞ is ample. r

Note that one cannot apply Weissauer’s theorem (Corollary 3.8) to the ring AfXg inorder to deduce that QuotðAfXgÞ is Hilbertian:

Proposition 5.3. For any subring A of Q, the ring AfXg is not a generalized Krull

domain.

Proof. Suppose R ¼ AfXg is a generalized Krull domain with essential family F.

Since the sumPyn¼1

2xn converges for all x A C with jxj < 1, the infinite product

f ðX Þ :¼ ð1 � 2XÞ � ð1 � 2X 2Þ � ð1 � 2X 3Þ � � �

converges on the open unit disc. Since it also converges X -adically, f A ZfXgLR. Foreach k A N, also fkðX Þ :¼ ð1 � 2X kÞ�1

f ðXÞ converges by the same argument, so fk A R,and hence 1 � 2X k divides f in R.

Since 1 � 2X k has a zero in the open unit disc and hence 1 � 2X k B R� we can fix avaluation vk A F with vkð1 � 2X kÞ > 0 for each k. If vk and vn are equivalent for somek < n, then vkð1 � 2X kÞ > 0 and vkð1 � 2X nÞ > 0 imply that

vkð2Þ þ vkðX kÞ þ vkð1 � X n�kÞ > 0:

But 1 � X n�k A R� and thus vkð2Þ > 0 or vkðXÞ > 0. However, since F is of finite type,this can happen for at most finitely many k.

We therefore get a cofinite subset NLN such that ðvkÞk AN is a family of pairwiseinequivalent valuations. Since 1 � 2X k divides f , we get that vkð f Þ > 0 for each k A N,contradicting the fact that F is of finite type. r

However, if A is a proper subring of Q, then A is contained in a rank-1 valuationring of Q ¼ QuotðAÞ (Fact 4.8), and hence QuotðAfXgÞ is Hilbertian by Proposition 4.3.Therefore, if in addition A3Z, then the above argument proves the inverse Galoisproblem for QuotðAfXgÞ. Combining this with Harbater’s result we get:

Theorem 5.4. Let A be a proper subring of Q. Then every finite group is a Galois

group over QuotðAfXgÞ.




Harbater’s result was recently generalized in a di¤erent direction in [Poi], Theo-reme 3.12, using the theory of analytic Berkovich spaces. We want to conclude by pointingout that Proposition 4.3 can also be used to easily reprove the positive answer to the inverseGalois problem for the rings considered in [Poi], Theoreme 3.14, and for many similarrings.

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Fachbereich Mathematik und Statistik, Universitat Konstanz, 78457 Konstanz, Germany

e-mail: [email protected]

The Open University of Israel, The Dorothy De Rothschild Campus, University Road 1,

Ra’anana 43104, Israel

e-mail: [email protected]

Eingegangen 10. Januar 2011, in revidierter Fassung 29. April 2011




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