A theorem of characterization of residual transcendental extensions of a valuation

Alexandru, Victor; Popescu, Nicolae; Zaharescu, Alexandru

doi:10.1215/kjm/1250520346

Cited by 73 publications

(55 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…i ∈ L and for every i the degree d i of |P| with respect to X i is less than n i , then by (47) and Theorem 2.1 from [1] Finally, we prove (48). Since the multiplicative group G/H, where G = |L × | L , H = |K × |, is generated by the imagesδ 1 , .…”

Section: Absolute Value On L(x) and There Exists A Finite Algebraic Ementioning

confidence: 70%

“…This problem has been affirmatively solved (see for example [1]). Some results on the corresponding problem for K(X 1 , .…”

mentioning

confidence: 83%

“…Without loss of generality we can consider P ∈ K[X] given by (1) such that all a j ∈B K (0, 1) and at least one of the coefficients has the absolute value equal to one. Since for every i, |X i | L = 1, it follows that…”

Section: Non-archimedean Absolute Values On K(x) In the Following Wementioning

confidence: 99%

“…Some problems connected with the norms on p-adic vector spaces were solved by I. S. Cohen [5] and A. F. Monna [8], and then O. Goldmann and N. Iwahori were concerned in [6] with the intrinsic structure that is carried by the set of all norms on a given finite dimensional vector space over a locally compact field. When r = 1, the case of K-algebra non-Archimedean norms on K [X] which are multiplicative and extend | | has been treated in [1][2][3]. In Section 2 below we consider generalizations of the Gauss valuation.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

ALL NON-ARCHIMEDEAN NORMS ON K[X₁, . . ., X_r]

Groza¹,

Popescu²,

Zaharescu³

2009

Glasgow Math. J.

Self Cite

View full text Add to dashboard Cite

Abstract. If K is a field with a non-trivial non-Archimedean absolute value (multiplicative norm) | |, we describe all non-Archimedean K-algebra norms on the polynomial algebra K[X 1 , . . . , X r ] which extend | |.2000 Mathematics Subject Classification. 11S75, 11C08. Introduction.Let K be a field with a non-trivial non-Archimedean absolute value (multiplicative norm) | |. In this paper, we study K-algebra non-Archimedean norms on K[X 1 , . . . , X r ] which extend | |. Some problems connected with the norms on p-adic vector spaces were solved by I. S. Cohen [5] and A. F. Monna [8], and then O. Goldmann and N. Iwahori were concerned in [6] with the intrinsic structure that is carried by the set of all norms on a given finite dimensional vector space over a locally compact field. When r = 1, the case of K-algebra non-Archimedean norms on K [X] which are multiplicative and extend | | has been treated in [1][2][3]. In Section 2 below we consider generalizations of the Gauss valuation. We investigate the case when a K-vector space norm is a K-algebra norm and we also address the question of when two norms are equivalent. In Section 3 we then discuss possible types of norms on K[X 1 , . . . , X r ] which extend a given non-trivial non-Archimedean absolute value on K. The completion of K[X 1 , . . . , X r ] with respect to a non-Archimedean Gauss norm is given in Section 4.There are many applications of non-Archimedean multiplicative norms on K[X 1 , . . . , X r ] in algebraic geometry where a basic tool is to describe all the absolute values on K(X 1 , . . . , X r ) which extend | |. In [7] F.-V. Kuhlmann determined which value groups and which residue fields can possibly occur in this case. In the case r = 1 the r.t. extensions | | L of | | to L = K(X) have been considered by M. Nagata [9], who

show abstract

Section: Absolute Value On L(x) and There Exists A Finite Algebraic Ementioning

confidence: 70%

“…This problem has been affirmatively solved (see for example [1]). Some results on the corresponding problem for K(X 1 , .…”

mentioning

confidence: 83%

Section: Non-archimedean Absolute Values On K(x) In the Following Wementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

ALL NON-ARCHIMEDEAN NORMS ON K[X₁, . . ., X_r]

Groza¹,

Popescu²,

Zaharescu³

2009

Glasgow Math. J.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Among the first papers describing valuations on rational function fields systematically were [M] and [MS]. Since then, an impressive number of papers have been written about the construction of such valuations and about their properties; the following list is by no means exhaustive: [AP], [APZ1]- [APZ3], [KH1]- [KH10], [KHG1]- [KHG6], [KHPR], [MO1], [MO2], [MOSW1], [O1]- [O3], [PP], [V]. From the paper [APZ3] the reader may get a good idea of how MacLane's original approach has been developed further.…”

Section: Introductionmentioning

confidence: 99%

Value groups, residue fields, and bad places of rational function fields

Kuhlmann

2004

Trans. Amer. Math. Soc.

113

View full text Add to dashboard Cite

Abstract. We classify all possible extensions of a valuation from a ground field K to a rational function field in one or several variables over K. We determine which value groups and residue fields can appear, and we show how to construct extensions having these value groups and residue fields. In particular, we give several constructions of extensions whose corresponding value group and residue field extensions are not finitely generated. In the case of a rational function field K(x) in one variable, we consider the relative algebraic closure of K in the henselization of K(x) with respect to the given extension, and we show that this can be any countably generated separablealgebraic extension of K. In the "tame case", we show how to determine this relative algebraic closure. Finally, we apply our methods to power series fields and the p-adics.

show abstract

Extensions of valuations to rational function fields over completions

Dutta

2023

Mathematische Nachrichten

View full text Add to dashboard Cite

Given a valued field (𝐾, 𝑣) and its completion ( K, 𝑣), we study the set of all possible extensions of 𝑣 to K(𝑋). We show that any such extension is closely connected with the underlying subextension (𝐾(𝑋)|𝐾, 𝑣). The connections between these extensions are studied via minimal pairs, key polynomials, pseudo-Cauchy sequences, and implicit constant fields. As a consequence, we obtain strong ramification theoretic properties of ( K, 𝑣). We also give necessary and sufficient conditions for (𝐾(𝑋), 𝑣) to be dense in ( K(𝑋), 𝑣).

show abstract

A theorem of characterization of residual transcendental extensions of a valuation

Cited by 73 publications

References 0 publications

ALL NON-ARCHIMEDEAN NORMS ON K[X₁, . . ., X_r]

ALL NON-ARCHIMEDEAN NORMS ON K[X₁, . . ., X_r]

Value groups, residue fields, and bad places of rational function fields

Extensions of valuations to rational function fields over completions

Contact Info

Product

Resources

About

A theorem of characterization of residual transcendental extensions of a valuation

Cited by 73 publications

References 0 publications

ALL NON-ARCHIMEDEAN NORMS ON K[X1, . . ., Xr]

ALL NON-ARCHIMEDEAN NORMS ON K[X1, . . ., Xr]

Value groups, residue fields, and bad places of rational function fields

Extensions of valuations to rational function fields over completions

Contact Info

Product

Resources

About

ALL NON-ARCHIMEDEAN NORMS ON K[X₁, . . ., X_r]

ALL NON-ARCHIMEDEAN NORMS ON K[X₁, . . ., X_r]