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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