Lemma 10.107.8. If $k \to S$ is an epimorphism and $k$ is a field, then $S = k$ or $S = 0$.

Proof. This is clear from the result of Lemma 10.107.7 (as any nonzero algebra over $k$ is faithfully flat), or by arguing directly that $R \to R \otimes _ k R$ cannot be surjective unless $\dim _ k(R) \leq 1$. $\square$

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