The Stacks project

Lemma 33.8.3. Let $X$ be a scheme over a separably closed field $k$. If $X$ is irreducible, then $X_ K$ is irreducible for any field extension $K/k$. I.e., $X$ is geometrically irreducible over $k$.

Proof. Use Properties, Lemma 28.3.3 and Algebra, Lemma 10.47.2. $\square$

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