Northcott property and universality of higher degree forms

Let [Formula: see text] be a totally real number field, [Formula: see text] a positive integer, and [Formula: see text] a higher degree form over [Formula: see text]. We prove that there are at most finitely many totally real extensions [Formula: see text] of degree [Formula: see text] such that [Formula: see text] over [Formula: see text] is universal. Further, we show that there are no universal forms over totally real infinite extensions of [Formula: see text] having the Northcott property.