Lemma 10.35.18. Let $R$ be a Jacobson ring. Let $K$ be a field. Let $R \subset K$ and $K$ is of finite type over $R$. Then $R$ is a field and $K/R$ is a finite field extension.

Proof. First note that $R$ is a domain. By Lemma 10.34.2 we see that $R_ f$ is a field and $K/R_ f$ is a finite field extension for some nonzero $f \in R$. Hence $(0)$ is a maximal ideal of $R_ f$ and by Lemma 10.35.14 we conclude $(0)$ is a maximal ideal of $R$. $\square$

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