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

1. for every maximal ideal $\mathfrak m$ of $S_ K$ the field $\kappa (\mathfrak m)$ is algebraic over $K$, and

2. $S_ K$ is a Jacobson ring.

Proof. Choose $k \subset K$ such that the cardinality of $K$ is greater than the cardinality of $S$. Since the elements of $S$ generate the $K$-algebra $S_ K$ we see that Theorem 10.35.11 applies. $\square$

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