  We show that if a universal quadratic form exists over an infinite degree,
totally real extension of the field of rationals $\mathbb{Q}$, then the set of
totally positive integers in the extension does not have the Northcott
property. In particular, this implies that no universal form exists over the
compositum of all totally real Galois fields of a fixed prime degree over
$\mathbb{Q}$. Further, by considering the existence of infinitely many square
classes of totally positive units, we show that no classical universal form
exists over the compositum of all such fields of degree $3d$ (for each fixed
odd integer $d$).
