On the closed subfields of \(C_p\)
type: Item
instance of: scholarly article
title: On the closed subfields of \(C_p\)
zbMATH Open document ID: 0901.11035
author: V. Alexandru
author: N. Popescu
author: Alexandru Zaharescu
published in: Journal of Number Theory
publication date: 1998-08-31T00:00:00Z
review text: Let \(p\) be a prime integer and \(\mathbb{Q}_p\) be the field of \(p\)-adic numbers and \(v\) the valuation of \(\mathbb{Q}_p\) characterised by \(v(p) =1\). Denote by \(\mathbb{Q}_p\) a fixed algebraic closure of \(\mathbb{Q}_p\) and also by \(v\) the unique extension of \(v\) to \(\overline {\mathbb{Q}}_p\). Let \((C_p,v)\) denote the completion of \((\mathbb{Q}_p,v)\). In an earlier paper \textit{A. Iovita} and \textit{A. Zaharescu} [cf. J. Number Theory 50, 202-205 (1995; Zbl 0813.12006)] proved that if \(K\) is a closed subfield of \(\mathbb{C}_p\) containing \(\mathbb{Q}_p\) which is not an algebraic extension of \(\mathbb{Q}_p\), then there exists \(t\) in \(K\) such that \(K\) is the completion of \(\mathbb{Q}_p (t)\). In this paper they show that \(t\) can be obtained as the limit of a certain well defined sequence. They introduce a particular kind of generic transcendental elements and give a criterion for two such elements to be conjugate. They have also shown that for \(t\in C_p\), \(\mathbb{Q}_p (t)\) and \(\mathbb{Q}_p [t]\) have the same topological closure.
Mathematics Subject Classification ID: 11S99
Mathematics Subject Classification ID: 12F05
zbMATH DE Number: 1121272
zbMATH Keywords: closed subfields of \(C_p\)
zbMATH Keywords: \(p\)-adic fields
zbMATH Keywords: algebraic extensions
zbMATH Keywords: generic transcendental elements
zbMATH Keywords: topological closure
Wikidata QID: Q105439581
reviewed by: Sudesh Kaur Khanduja
MaRDI profile type: Publication
full work available at URL: jnth.1997.2198
OpenAlex ID: W2005515483
cites work: Completion of r.t. extensions of local fields. I
cites work: A theorem of characterization of residual transcendental extensions of a valuation
cites work: Minimal pairs of definition of a residual transcendental extension of a valuation
cites work: All valuations on K(X)
cites work: Some elementary remarks about \(n\)-local fields
cites work: Q5590141
cites work: Zeros of polynomials over local fields. The Galois action
cites work: Completions of r.a.t.-valued fields of rational functions
cites work: Nondiscrete local ramified class field theory
cites work: Q3475353
cites work: On the residual transcendental extensions of a valuation. Key polynomials and augmented valuation
cites work: On the structure of the irreducible polynomials over local fields
cites work: Q4342937
cites work: Q5509718
cites work: Q5723265
DOI: 10.1006/JNTH.1997.2198
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description: scientific article; zbMATH DE number 1121272