On the Northcott property and other properties related to polynomial mappings

2011/11/21 by Sara Checcoli, Checcoli, Sara, Martin Widmer +1
Mathematics · #11G50 #11R04 (Primary) 37P05 (Secondary) #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11G50 #msc:11R04 #msc:37P05

paper · pdf · doi:10.48550/arxiv.1111.5060

14 pages

arxiv created 2011/11/21 · openalex publication_date 2011/11/21 · arxiv updated 2011/11/23 · openalex created_date 2022/09/25 · openalex updated_date 2026/07/28

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 \G_m\_m there exists a sequence of Galois extensions \K_m\_m with Gal(K_m/ℚ)=G_m such that the compositum of the fields K_m 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.

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