Monatshefte für Mathematik

, Volume 162, Issue 3, pp 341–353 | Cite as

On certain infinite extensions of the rationals with Northcott property



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 \({{\mathbb Q}_{ab}^{(d)}}\) , 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. \({{\mathbb Q}(2^{1/d_1},3^{1/d_2},5^{1/d_3},7^{1/d_4},11^{1/d_5},\ldots)}\) has the Northcott property if and only if \({2^{1/d_1}, 3^{1/d_2}, 5^{1/d_3}, 7^{1/d_4}, 11^{1/d_5}}\) , . . . tends to infinity.


Northcott property Height Preperiodic points Field arithmetic 

Mathematics Subject Classification (2000)

12F05 11G50 


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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Texas at AustinAustinUSA
  2. 2.Institut für Mathematik ATechnische Universität GrazGrazAustria

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