Lemma 10.36.19. Let $k$ be a field. Let $S$ be a $k$-algebra over $k$.

If $S$ is a domain and finite dimensional over $k$, then $S$ is a field.

If $S$ is integral over $k$ and a domain, then $S$ is a field.

If $S$ is integral over $k$ then every prime of $S$ is a maximal ideal (see Lemma 10.26.5 for more consequences).

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