The Stacks project

Lemma 10.47.5. Let $k$ be a field. Let $R$ be a $k$-algebra. If $k$ is separably algebraically closed then $R$ is geometrically irreducible over $k$ if and only if the spectrum of $R$ is irreducible.

Proof. Immediate from the remark following Definition 10.47.4. $\square$

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