On the Northcott property and other properties related to polynomial mappings

Abstract

We prove that if K/Q is a Galois extension of finite exponent andis 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.

Two remarks on Narkiewicz's property (P)

“…An algebraic extension F of Q is said to have the Northcott property (N) if for every T ∈ R the set {α ∈ F |h(α) ≤ T } is finite. It is easy to see that property (N) implies property (P), we refer to [4] and [6] for a proof and additional results. There are only few examples known of subfields of Q with property (P).…”

Undecidability, unit groups, and some totally imaginary infinite extensions of $\mathbb{Q}$

“…The following theorem of Bombieri and Zannier provides many examples of fields with the Northcott property; see also [4,30] for more examples. By definition, it suffices to show that the totally real field K is contained in a (possibly imaginary) field which has the Northcott property.…”

Les Huit Premiers Travaux de Pierre Liardet

“…Ses résultats ontété appliqués récemment pourétudier des ensembles ayant la propriété de Northcott, notamment par S. Checcoli et M. Widmer [21,39].…”