The Stacks project

Lemma 33.25.10. Let $k$ be a field. Let $X$ be a variety over $k$ which has a $k$-rational point $x$ such that $X$ is smooth at $x$. Then $X$ is geometrically integral over $k$.

Proof. Let $U \subset X$ be the smooth locus of $X$. By assumption $U$ is nonempty and hence dense and scheme theoretically dense. Then $U_{\overline{k}} \subset X_{\overline{k}}$ is dense and scheme theoretically dense as well (some details omitted). Thus it suffices to show that $U$ is geometrically integral. Because $U$ has a $k$-rational point it is geometrically connected by Lemma 33.7.14. On the other hand, $U_{\overline{k}}$ is reduced and normal (Lemma 33.25.4. Since a connected normal Noetherian scheme is integral (Properties, Lemma 28.7.6) the proof is complete. $\square$

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