Extensions de valuation et polygone de Newton
type: Item
instance of: scholarly article
title: Extensions de valuation et polygone de Newton
zbMATH Open document ID: 1170.13003
author: Michel Vaquié
publication date: 2009-02-10T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/10385
review text: Let \((K,\nu )\) be a valued field and \(L=K(\theta)\) a finite cyclic extension of \(K\). The aim of this paper is to determine all valuations extending \(\nu\). \(L\) is isomorphic to \(K[x]/(P)\), and any valuation of \(L\) which extends \(\nu \) defines a pseudo-valuation \(\zeta \) on \(K[x]\) whose kernel is the principal ideal \((P)\). The author associates to \(\zeta \) a family of valuations on \(K[x]\), called an admissible family, which is explicitly constructed by augmented valuations and limit augmented valuations. The author gives a necessary and sufficient condition for a valuation of \(K[x]\) to belong to an admissible family associated to a pseudo-valuation \(\zeta \) which corresponds to a valuation of \(L\), this condition depends only on the polynomial \(P\). The author determines all the valuations of \(L\) which extend the valuation \(\nu \) of \(K\). In order to solve this problem the author defines the Newton polygon \({\mathcal PN}(P;\varphi;\mu)\), associated to \(P\), to a polynomial \(\varphi \) and to a valuation \(\mu \) of \(K[x]\).
reviewed by: Marcel Morales
Mathematics Subject Classification ID: 13A18
Mathematics Subject Classification ID: 12J10
Mathematics Subject Classification ID: 14E15
zbMATH DE Number: 5505490
zbMATH Keywords: valuation
zbMATH Keywords: extension
zbMATH Keywords: Newton polygon
MaRDI profile type: Publication
OpenAlex ID: W2333763798
cites work: A construction for prime ideals as absolute values of an algebraic field
cites work: Anneaux locaux henséliens
cites work: Q4945017
cites work: Q5708415
cites work: Q3596115
cites work: Extension d’une valuation
cites work: Admissible family of valuations and defect of an extension.
DOI: 10.5802/AIF.2421
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 5505490