On a local property of infinite Galois extensions implying the Northcott property

TL;DR

This paper investigates a local property of infinite Galois extensions related to the Northcott property, showing it is not preserved under finite extensions and cannot be deduced solely from the Galois group structure.

Contribution

It demonstrates that the local property is not invariant under finite extensions and cannot be inferred from the Galois group, contrasting with previous assumptions.

Findings

Local property not preserved under finite extensions

Local property cannot be deduced from Galois group structure

Existence of fields with Galois groups not satisfying the local property

Abstract

In 2001, Bombieri and Zannier studied the Northcott property (N) for infinite Galois extensions of the rationals. In particular they provided a local property of the extensions that imply property (N). Later, Checcoli and Fehm demonstrated the existence of infinite extensions satisfying this local property. In this article, we establish two main results. First, we show that this local property, unlike property (N), is not preserved under finite extensions. Second, we show that, for an infinite Galois extension of Q, such local property cannot be read on the Galois group. More precisely, we exhibit several profinite groups that are realizable over Q by fields that do not satisfy the local property.