On certain infinite extensions of the rationals with Northcott property

TL;DR

This paper investigates infinite algebraic extensions of the rationals with the Northcott property, providing a simple criterion and new examples of such fields, including certain infinite radical extensions.

Contribution

It introduces a straightforward criterion for the Northcott property and applies it to generate new examples of infinite degree fields with this property.

Findings

Fields generated by sequences tending to infinity have the Northcott property.

$ ext{Q}(2^{1/d_1},3^{1/d_2},...,p^{1/d_k})$ has the Northcott property if the sequence tends to infinity.

Provides new examples of infinite algebraic extensions with the Northcott property.

Abstract

A set of algebraic numbers has the Northcott property if each of its subsets of bounded Weil height is finite. Northcott's Theorem, which has many Diophantine applications, states that sets of bounded degree have the Northcott property. Bombieri, Dvornicich and Zannier raised the problem of finding fields of infinite degree with this property. Bombieri and Zannier have shown that $\IQ_{ab}^{(d)}$, the maximal abelian subfield of the field generated by all algebraic numbers of degree at most $d$, is such a field. In this note we give a simple criterion for the Northcott property and, as an application, we deduce several new examples, e.g. $\IQ(2^{1/d_1},3^{1/d_2},5^{1/d_3},7^{1/d_4},11^{1/d_5},...)$ has the Northcott property if and only if $2^{1/d_1},3^{1/d_2},5^{1/d_3},7^{1/d_4},11^{1/d_5},...$ tends to infinity.