Most totally real fields do not have universal forms or the Northcott property.

Daans, Nicolas; Kala, Vítězslav; Man, Siu Hang; Widmer, Martin; Yatsyna, Pavlo · Proc Natl Acad Sci U S A · 2025

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Abstract

We show that, in the space of all totally real fields equipped with the constructible topology, the set of fields that admit a universal quadratic form, or have the Northcott property, is meager. The main tool is a theorem on the number of square classes of totally positive units represented by a quadratic lattice of a given rank.