On the Northcott property and local degrees

TL;DR

This paper constructs specific infinite Galois extensions of the rationals with the Northcott property based on prime splitting behavior, and provides examples of extensions lacking this property despite finite local degrees.

Contribution

It introduces new classes of Galois extensions with the Northcott property derived from prime splitting, and shows extensions with finite local degrees that do not satisfy the property.

Findings

Constructed infinite Galois extensions with the Northcott property from prime splitting behavior.

Provided examples of extensions with finite local degrees that lack the Northcott property.

Abstract

We construct infinite Galois extensions $K$ of $\mathbb{Q}$ that satisfy the Northcott property on elements of small height, and where this property can be deduced solely from the splitting behavior of prime numbers in $K$. We also give examples of Galois extensions of $\mathbb{Q}$ which have finite local degree at all prime numbers and do not satisfy the Northcott property.