Lifting problem for universal quadratic forms

We study totally real number fields that admit a universal quadratic form whose coefficients are rational integers. We show that Q(root 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 Q(zeta(7) + zeta(-1)(7)), 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. (C) 2020 Elsevier Inc. All rights reserved.