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On the Northcott property and other properties related to polynomial mappings

Published online by Cambridge University Press:  27 February 2013

SARA CHECCOLI  and
MARTIN WIDMER
SARA CHECCOLI
Affiliation:
Institute of Mathematics, University of Basel, Rheinsprung 21, CH-4051 Basel, Switzerland. e-mail: [email protected]
MARTIN WIDMER
Affiliation:
Department for Analysis and Computational Number Theory, Graz University of Technology, Steyrergasse 30/II, 8010 Graz, Austria. e-mail: [email protected]
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Abstract

We prove that if K/ℚ is a Galois extension of finite exponent and K(d) is the compositum of all extensions of K of degree at most d, then K(d) has the Bogomolov property and the maximal abelian subextension of K(d)/ℚ has the Northcott property.

Moreover, we prove that given any sequence of finite solvable groups {Gm}m there exists a sequence of Galois extensions {Km}m with Gal(Km/ℚ)=Gm such that the compositum of the fields Km has the Northcott property. In particular we provide examples of fields with the Northcott property with uniformly bounded local degrees but not contained in ℚ(d).

We also discuss some problems related to properties introduced by Liardet and Narkiewicz to study polynomial mappings. Using results on the Northcott property and a result by Dvornicich and Zannier we easily deduce answers to some open problems proposed by Narkiewicz.

Keywords

Primary 11G5011R04Secondary 37P05
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 155 , Issue 1 , July 2013 , pp. 1 - 12
Copyright
Copyright © Cambridge Philosophical Society 2013 

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