  Let $K$ be a totally real number field, $d$ a positive integer, and $Q$ a
higher degree form over $K$. We prove that there are at most finitely many
totally real extensions $L/K$ of degree $d$ such that $Q$ over $L$ is
universal. Further, we show that there are no universal forms over totally real
infinite extensions of $\mathbb{Q}$ having the Northcott property.
