Lemma 10.36.18. 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 10.36.17 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 10.25.1). The other assertions are immediate from this. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 00GR. Beware of the difference between the letter 'O' and the digit '0'.