Three counterexamples concerning the Northcott property of fields

TL;DR

This paper presents three distinct examples of fields related to the Northcott property, illustrating nuanced differences in algebraic and arithmetic properties such as Galois closure, pseudo-algebraic closure, and local degrees.

Contribution

It introduces three novel counterexamples that clarify the boundaries and relationships between the Northcott property, Bogomolov property, and other field characteristics.

Findings

First example: Northcott property holds but Galois closure lacks Bogomolov property.

Second example: Northcott property and pseudo-algebraic closure coexist.

Third example: Bounded local degrees at infinitely many primes without Northcott property.

Abstract

We give three examples of fields concerning the Northcott property on elements of small height: The first one has the Northcott property but its Galois closure does not satisfy the Bogomolov property. The second one has the Northcott property and is pseudo-algebraically closed, i.e. every variety has a dense set of rational points. The third one has bounded local degree at infinitely many rational primes but does not have the Northcott property.