Lemma 29.36.8. Let f : X \to S be a morphism of schemes. If f is flat, locally of finite presentation, and for every s \in S the fibre X_ s is a disjoint union of spectra of finite separable field extensions of \kappa (s), then f is étale.
Proof. You can deduce this from Algebra, Lemma 10.143.7. Here is another proof.
By Lemma 29.36.7 a fibre X_ s is étale and hence smooth over s. By Lemma 29.34.3 we see that X \to S is smooth. By Lemma 29.35.12 we see that f is unramified. We conclude by Lemma 29.36.5. \square
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