On the Northcott property and other properties related to polynomial mappings

TL;DR

This paper investigates properties of polynomial mappings related to the Northcott and Bogomolov properties, proving new results about the structure of certain Galois extensions and providing examples with bounded local degrees.

Contribution

It establishes that certain compositums of Galois extensions have the Northcott and Bogomolov properties, and constructs examples of fields with these properties that are not contained in known classes.

Findings

Compositum of all degree ≤ d extensions of a Galois extension has the Bogomolov property.

Maximal abelian subextensions of these compositums have the Northcott property.

Constructed Galois extensions with prescribed Galois groups whose compositum has the Northcott property.

Abstract

We prove that if $K/\mathbb{Q}$ 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 $K^{(d)}/\mathbb{Q}$ 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 $\text{Gal}(K_m/\mathbb{Q})=G_m$ 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 $\mathbb{Q}^{(d)}$. 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.