Universal quadratic forms and Northcott property of infinite number fields

TL;DR

The paper investigates the existence of universal quadratic forms over infinite totally real extensions of ield and shows such forms cannot exist if the set of totally positive integers lacks the Northcott property, with implications for various field compositums.

Contribution

It establishes new non-existence results for universal quadratic forms over infinite totally real fields, linking their existence to the Northcott property and properties of totally positive units.

Findings

Universal quadratic forms do not exist over certain infinite totally real extensions.

The set of totally positive integers in these extensions lacks the Northcott property.

No classical universal form exists over the compositum of all totally real Galois fields of fixed prime degree.

Abstract

We show that if a universal quadratic form exists over an infinite degree, totally real extension of the field of rationals $\mathbb{Q}$, then the set of totally positive integers in the extension does not have the Northcott property. In particular, this implies that no universal form exists over the compositum of all totally real Galois fields of a fixed prime degree over $\mathbb{Q}$. Further, by considering the existence of infinitely many square classes of totally positive units, we show that no classical universal form exists over the compositum of all such fields of degree $3d$ (for each fixed odd integer $d$).