On certain infinite extensions of the rationals with Northcott property View Full Text


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Article Info

DATE

2009-10-23

AUTHORS

Martin Widmer

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

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

    DOI

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

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