Lifting problem for universal quadratic forms

We study totally real number fields that admit a universal quadratic form whose coefficients are rational integers. show Q(5) is the only such field, and among of degrees 3, 4, 5, 7 which have principal codifferent ideal, one Q(ζ7+ζ7−1), over x2+y2+z2+w2+xy+xz+xw universal. Moreover, we prove an upper bound for Pythagoras numbers orders in depends on degree field.