Lifting problem for universal quadratic forms

TL;DR

This paper characterizes real quadratic fields with universal quadratic forms, identifies a unique higher-degree example, and provides bounds on Pythagoras numbers based on the degree of number fields.

Contribution

It proves that only Q(√5) admits a universal quadratic form with integer coefficients among real quadratic fields and identifies a unique degree 7 example, also establishing bounds on Pythagoras numbers.

Findings

Q(√5) is the only real quadratic field with a universal quadratic form.

Q(ζ_7 + ζ_7^{-1}) is the only degree 7 field with principal codifferent ideal and a universal form.

Upper bounds for Pythagoras numbers depend solely on the degree of the number field.

Abstract

We study totally real number fields that admit a universal quadratic form whose coefficients are rational integers. We show that $\mathbb Q(\sqrt 5)$ is the only such real quadratic field, and that among fields of degrees 3, 4, 5, and 7 which have principal codifferent ideal, the only one is $\mathbb Q(\zeta_7+\zeta_7^{-1})$, over which the form $x^2+y^2+z^2+w^2+xy+xz+xw$ is universal. Moreover, we prove an upper bound for Pythagoras numbers of orders in number fields that depends only on the degree of the number field.