Lemma 10.35.12. Let $k$ be a field. Let $S$ be a $k$-algebra. For any field extension $K/k$ whose cardinality is larger than the cardinality of $S$ we have
for every maximal ideal $\mathfrak m$ of $S_ K$ the field $\kappa (\mathfrak m)$ is algebraic over $K$, and
$S_ K$ is a Jacobson ring.