Norms on \(F(X)\)
type: Item
instance of: scholarly article
title: Norms on \(F(X)\)
zbMATH Open document ID: 0491.12024
published in: Pacific Journal of Mathematics
publication date: 1983-01-01T00:00:00Z
review text: It is well known that if \(\Vert\cdot\Vert\) is a norm on the field \(F(X)\) of rational functions over a field \(F\) for which \(F\) is bounded, then \(\Vert\cdot\Vert\) is equivalent to the supremum of a finite family of absolute values on \(F(X)\), each of which is improper on \(F\). Moreover, \(\Vert\cdot\Vert\) is equivalent to an absolute value if and only if the completion of \(F(X)\) for \(\Vert\cdot\Vert\) is a field. We show that the analogous characterization of norms on \(F(X)\) for which \(F\) is discrete is impossible by constructing for each infinite field \(F\), a norm \(\Vert\cdot\Vert\) on \(F(X)\) for which \(F\) is discrete, \(\Vert X\Vert <1\), the completion of \(F(X)\) for \(\Vert\cdot\Vert\) is a field but \(\Vert\cdot\Vert\) is not equivalent to the supremum of finitely many absolute values.
Mathematics Subject Classification ID: 12J05
Mathematics Subject Classification ID: 12J10
zbMATH DE Number: 3771760
zbMATH Keywords: rational function field
zbMATH Keywords: absolute values
zbMATH Keywords: bounded norm
zbMATH Keywords: valuation
zbMATH Keywords: completion
author: Jo-Ann D. Cohen
MaRDI profile type: Publication
full work available at URL: pjm.1983.109.81
OpenAlex ID: W1963696867
DOI: 10.2140/PJM.1983.109.81
description: scientific article; zbMATH DE number 3771760