Lemma 10.35.18. Let $R$ be a ring. Let $K$ be a field. If $R \subset K$ and $K$ is integral over $R$, then $R$ is a field and $K$ is an algebraic extension. If $R \subset K$ and $K$ is finite over $R$, then $R$ is a field and $K$ is a finite algebraic extension.

**Proof.**
Assume that $R \subset K$ is integral. By Lemma 10.35.17 we see that $\mathop{\mathrm{Spec}}(R)$ has $1$ point. Since clearly $R$ is a domain we see that $R = R_{(0)}$ is a field (Lemma 10.24.1). The other assertions are immediate from this.
$\square$

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