On the properties of Northcott and of Narkiewicz for fields of algebraic numbers
type: Item
instance of: scholarly article
title: On the properties of Northcott and of Narkiewicz for fields of algebraic numbers
zbMATH Open document ID: 1186.12002
published in: Functiones et Approximatio. Commentarii Mathematici
publication date: 2009-11-18T00:00:00Z
review text: A subfield \(F\) of \(\overline {\mathbb Q}\), the algebraic closure of the rational field, is said to have the Northcott property, if for every \(B>0\) it contains only finitely many elements \(x\) with \(h(x)\leq B\), \(h(x)\) being Weil's logarithmic height [\textit{D. G. Northcott}, Ann. Math. (2) 51, 167--177 (1950; Zbl 0036.30102)]. The field \(F\) has the property \((P)\), if every subset \(A\subset F\) for which there exists a nonlinear polynomial \(f\in F(X)\) with \(f(A)=A\) must be finite [the reviewer, Acta Arith. 7, 241--249, ibid. 8, 11--19 (1962; Zbl 0125.00901)]. The authors give a survey of recent results dealing with these properties and prove that these two properties are not equivalent.
reviewed by: Władysław Narkiewicz
Mathematics Subject Classification ID: 12E05
Mathematics Subject Classification ID: 11R04
Mathematics Subject Classification ID: 11G50
zbMATH DE Number: 5635545
zbMATH Keywords: Weil logarithmic height
zbMATH Keywords: polynomial mappings
zbMATH Keywords: preperiodic points
zbMATH Keywords: Northcott property
author: Roberto Dvornicich
author: Umberto Zannier
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.7169/facm/1229696562
OpenAlex ID: W1971217875
DOI: 10.7169/FACM/1229696562
description: scientific article; zbMATH DE number 5635545