The algebraic closure of a formal power series field
type: Item
instance of: scholarly article
title: The algebraic closure of a formal power series field
zbMATH Open document ID: 0849.12015
author: Ali Benhissi
published in: Annales Mathématiques Blaise Pascal
publication date: 1996-11-11T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/79136
review text: This paper is devoted to the study of the algebraic closure \(\overline {k((T) )}\) of the field of meromorphic formal power series \(k((T))\) in one indeterminate over a field \(k\) of characteristic \(p \neq 0\). It is proved that this algebraic closure is the limit of an increasing sequence of Galois extensions starting from the Puiseux field \(k((T^{{1 \over \infty}})) = \cup_{n = 1}^\infty k((T^{{1 \over n}}))\). The separable closure of \(k((T))\) is also described. We note that the paper contains several interesting examples. Some of them are based on techniques developed by \textit{F. J. Rayner} [Glasg. Math. J. 9, 146-151 (1968; Zbl 0159.05302)], \textit{P. Ribenboim} and \textit{L. Van der Vries} [Can. Math. Bull. 27, 313-315 (1984; Zbl 0548.12013)] and \textit{D. Ştefănescu} [Bull. Math. Soc. Sci. Math. Repub. Soc. Roum., Nouv. Ser. 26, 77-91 (1982; Zbl 0487.12015), ibid. 27, 169-178 (1983; Zbl 0515.12018)]. The author proves that there exist two generalized power series with the same support but such that one is algebraic over \(k((T))\) and the other is transcendental.
Mathematics Subject Classification ID: 12J15
Mathematics Subject Classification ID: 12J10
zbMATH DE Number: 871696
zbMATH Keywords: Galois extensions of the Puiseux field
zbMATH Keywords: separable closure
zbMATH Keywords: algebraic closure
zbMATH Keywords: meromorphic formal power series
reviewed by: Doru Ştefănescu
MaRDI profile type: Publication
OpenAlex ID: W2333850340
cites work: Q5653950
cites work: Commutative rings whose finitely generated modules decompose
cites work: An algebraically closed field
cites work: Algebraically Closed Fields Analogous to Fields of Puiseux Series
cites work: Q5666659
cites work: The Absolute Galois Group of a Rational Function Field in Characteristic Zero is a Semi-Direct Product
cites work: Q3947756
cites work: Q3661690
cites work: Q5732721
DOI: 10.5802/AMBP.42
Recommended article: The algebraic closure of the power series field in positive characteristic
Recommended article: About the algebraic closure of the field of power series in several variables in characteristic zero
Recommended article: Q4327035
Recommended article: Power series and \(p\)-adic algebraic closures
Recommended article: Q3499961
Recommended article: Q3356398
Recommended article: A Note on the Algebraic Closure of a Field
Recommended article: \(t\)-closed rings of formal power series
Recommended article: Algebraic elements in formal power series rings
Recommended article: Q4957926
description: scientific article; zbMATH DE number 871696