Lifting problem for universal quadratic forms

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.

16 pages, incorporated referee comments