  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.
