Description of the étale schemes over fields and fibres of étale morphisms.
Lemma 29.36.7. Fibres of étale morphisms.
Let $X$ be a scheme over a field $k$. The structure morphism $X \to \mathop{\mathrm{Spec}}(k)$ is étale if and only if $X$ is a disjoint union of spectra of finite separable field extensions of $k$.
If $f : X \to S$ is an étale morphism, then for every $s \in S$ the fibre $X_ s$ is a disjoint union of spectra of finite separable field extensions of $\kappa (s)$.
Proof. You can deduce this from Lemma 29.35.11 via Lemma 29.36.5 above. Here is a direct proof.
We will use Algebra, Lemma 10.143.4. Hence it is clear that if $X$ is a disjoint union of spectra of finite separable field extensions of $k$ then $X \to \mathop{\mathrm{Spec}}(k)$ is étale. Conversely, suppose that $X \to \mathop{\mathrm{Spec}}(k)$ is étale. Then for any affine open $U \subset X$ we see that $U$ is a finite disjoint union of spectra of finite separable field extensions of $k$. Hence all points of $X$ are closed points (see Lemma 29.20.2 for example). Thus $X$ is a discrete space and we win. $\square$
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