Universality lifting from a general base field

TL;DR

This paper proves finiteness results for the number of totally real extensions of a fixed degree over a base field that admit universal quadratic forms, providing explicit classifications for certain cases.

Contribution

It establishes new finiteness theorems for universal quadratic forms over totally real extensions and offers explicit classifications for relative quadratic extensions.

Findings

Finitely many totally real extensions admit universal quadratic forms.

Explicit classifications obtained for certain relative quadratic extensions.

New finiteness results in the theory of universal quadratic forms.

Abstract

Given a totally real number field $F$, we show that there are only finitely many totally real extensions of $K$ of a fixed degree that admit a universal quadratic form defined over $F$. We further obtain several explicit classification results in the case of relative quadratic extensions.