# On certain infinite extensions of the rationals with Northcott property

Ontology type: schema:ScholarlyArticle      Open Access: True

### Article Info

DATE

2009-10-23

AUTHORS ABSTRACT

A set of algebraic numbers has the Northcott property if each of its subsets of bounded Weil height is finite. Northcott’s Theorem, which has many Diophantine applications, states that sets of bounded degree have the Northcott property. Bombieri, Dvornicich and Zannier raised the problem of finding fields of infinite degree with this property. Bombieri and Zannier have shown that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb Q}_{ab}^{(d)}}$$\end{document} , the maximal abelian subfield of the field generated by all algebraic numbers of degree at most d, is such a field. In this note we give a simple criterion for the Northcott property and, as an application, we deduce several new examples, e.g. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb Q}(2^{1/d_1},3^{1/d_2},5^{1/d_3},7^{1/d_4},11^{1/d_5},\ldots)}$$\end{document} has the Northcott property if and only if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${2^{1/d_1}, 3^{1/d_2}, 5^{1/d_3}, 7^{1/d_4}, 11^{1/d_5}}$$\end{document} , . . . tends to infinity. More... »

PAGES

341-353

### References to SciGraph publications

• 1977. Number Fields in NONE
• 1999. Algebraic Number Theory in NONE
• 1998. The Theory of Algebraic Number Fields in NONE
• 2001-03. SUSY long-lived massive particles: Detection and physics at the LHC in RENDICONTI LINCEI. SCIENZE FISICHE E NATURALI
• 1994. Algebraic Number Theory in NONE
• ### Journal

TITLE

Monatshefte für Mathematik

ISSUE

3

VOLUME

162

### Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s00605-009-0162-7

DOI

http://dx.doi.org/10.1007/s00605-009-0162-7

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1041865150

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