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 Q(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(ζ7+ζ7−1), over which the form x2+y2+z2+w2+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.