Minimal pairs, minimal fields and implicit constant fields

Dutta, Arpan

doi:10.1016/j.jalgebra.2021.09.008

Cited by 8 publications

(19 citation statements)

References 17 publications

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“…It follows from Theorem 4.2 that {a ν } ν<λ is a Cauchy sequence in (K, v) and the unique limit a ∈ K is algebraic over K. Moreover, (a, γ) is the unique pair of definition for the induced extension ( K(X)| K, v). It then follows from [7,Theorem 7.2…”

Section: Density Of K(x) In K(x)mentioning

confidence: 83%

“…Since we chose a ν to be separable-algebraic over K, the extension K h (a ν | ν < λ)| K h is also separable-algebraic. It now follows from [7,Theorem 7.2] that…”

Section: Density Of K(x) In K(x)mentioning

confidence: 96%

“…The assertion that ( K(X)|K(X), v) is an immediate extension now follows from [7, Remark 3.1] and [7,Remark 3.3].…”

Section: Vk Is Cofinal In Vk(x)mentioning

confidence: 98%

“…Assume that ( K(X)|K, v) is an extension of valued fields such that ( K(X)| K, v) is value transcendental with a unique pair of definition (a, γ). It follows from [7,Remark 3.3] that v K(X) = Zγ ⊕ vK. Further, the fact that (a, γ) is the unique pair of definition implies that γ > vK.…”

Section: Observe That ( K(x)| K V Aν γνmentioning

confidence: 99%

“…A thorough understanding of this extension problem has given rise to a whole host of literature and several different inter-connected notions. The fundamental objects of study are minimal pairs of definition [1,4,2,3], key polynomials [16,20,18,17], pseudo-Cauchy sequences [19,9] and implicit constant fields [14,7,6,8]. We refer the reader to Section 2 for all the relevant definitions.…”

Section: Introductionmentioning

confidence: 99%

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Extensions of valuations to rational function fields over completions

Dutta¹

2022

Preprint

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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).In this article, we undertake a systematic study of the extension ( K(X)| K, v), where K denotes the completion of the valued field (K, v). Our investigation shows that the extension ( K(X)| K, v) is intimately connected with the underlying extension (K(X) |K, v). The connection between these extensions is further explored in detail in terms of the aforementioned objects, viz. minimal pairs, key polynomials and implicit constant fields. Our first important result is the following: Theorem 1.1. Let (K, v) be a valued field, ( K, v) be its completion and (K(X)|K, w) be an extension of valued fields. Fix an extension v of v to the algebraic closure K of K. Fix a common extension w of w and v to K(X), where v := v| K . Then there exists a common extension of v

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Section: Density Of K(x) In K(x)mentioning

confidence: 83%

“…Since we chose a ν to be separable-algebraic over K, the extension K h (a ν | ν < λ)| K h is also separable-algebraic. It now follows from [7,Theorem 7.2] that…”

Section: Density Of K(x) In K(x)mentioning

confidence: 96%

“…The assertion that ( K(X)|K(X), v) is an immediate extension now follows from [7, Remark 3.1] and [7,Remark 3.3].…”

Section: Vk Is Cofinal In Vk(x)mentioning

confidence: 98%

Section: Observe That ( K(x)| K V Aν γνmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Extensions of valuations to rational function fields over completions

Dutta¹

2022

Preprint

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Extensions of valuations to rational function fields over completions

Dutta

2023

Mathematische Nachrichten

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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).

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An invariant of valuation transcendental extensions and its connection with key polynomials

Dutta

2024

Journal of Algebra

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Minimal pairs, minimal fields and implicit constant fields

Cited by 8 publications

References 17 publications

Extensions of valuations to rational function fields over completions

Extensions of valuations to rational function fields over completions

Extensions of valuations to rational function fields over completions

An invariant of valuation transcendental extensions and its connection with key polynomials

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