Norms on \(R[ X_{1}, \dots , X_r]\) which are multiplicative on \(R\)
type: Item
instance of: scholarly article
title: Norms on \(R[ X_{1}, \dots , X_r]\) which are multiplicative on \(R\)
zbMATH Open document ID: 1140.11055
author: Ghiocel Groza
author: N. Popescu
author: Alexandru Zaharescu
published in: Results in Mathematics
publication date: 2008-06-10T00:00:00Z
review text: \loadeufm The main motivation of the paper under review comes from the theory of local fields. By Galois theory in \({\mathbb C}_ p\), the completion of an algebraic closure \(\overline{\mathbb Q}_ p\) of the field of \(p\)-adic numbers \({\mathbb Q}_ p\), there is a natural correspondence between the closed subgroups of the Galois groups \(\roman{Gal}_{\text{cont}}({\mathbb C}_ p/{\mathbb Q}_ p)\) of continuous automorphisms of \({\mathbb C}_ p\) over \({\mathbb Q}_ p\) and the closed subfields of \({\mathbb C}_ p\). Now, for each closed subfield \(E\) of \({\mathbb C}_ p\), there exists an element \(T\) of \(E\) such that \(E\) is the topological closure of the ring \({\mathbb Q}_ p[T]\) in \({\mathbb C}_ p\), where \(T\) is transcendental over \({\mathbb Q}_ p\) when \(E\) is not a finite extension of \({\mathbb Q}_ p\). Although all these rings are ring isomorphic, what is very different for various transcendental elements \(T\) over \({\mathbb Q}_ p\) is the valuation on \({\mathbb C}_ p\) restricted to \({\mathbb Q}_ p[T]\). That is, this variety of closed subfields of \({\mathbb C}_ p\) is determined by these extensions of the \(p\)-adic valuation from \({\mathbb Q}_ p\) to the ring \({\mathbb Q}_ p[X]\) of polynomials of one variable over \({\mathbb Q}_ p\). There are some other objects obtained in a similar way: the rings \(B_{dR}^ +\) introduced by \textit{J.-M. Fontaine} [Ann. Math. (2) 115, 529--577 (1982; Zbl 0544.14016)], the rings \(B_ n:=B_{dR}^ +/I^ n\) where \(I\) is the maximal ideal of \(B_{dR}^ +\), have generating elements over \({\mathbb Q}_ p\), so these rings can also be obtained as completions of \({\mathbb Q}_ p[X]\) with respect to some norms which extend the \(p\)-adic absolute value from \({\mathbb Q}_ p\). Other motivation comes from the structure of some spaces of functions, which were introduced to define some types of convergence for different sequences of real functions, which are not uniformly convergent. These spaces are completi…
Mathematics Subject Classification ID: 11S15
Mathematics Subject Classification ID: 11C08
zbMATH DE Number: 5285866
zbMATH Keywords: polynomial algebras
zbMATH Keywords: norm extensions
zbMATH Keywords: local fields
zbMATH Keywords: function fields
reviewed by: Gabriel D. Villa Salvador
MaRDI profile type: Publication
full work available at URL: s00025-007-0273-1
OpenAlex ID: W2034864043
DOI: 10.1007/S00025-007-0273-1
description: scientific article; zbMATH DE number 5285866