The Stacks project

Lemma 72.14.2. Let $k$ be a field. Let $X$ be a decent algebraic space over $k$. Then $X$ is geometrically integral over $k$ if and only if $X$ is both geometrically reduced and geometrically irreducible over $k$.

Proof. This is an immediate consequence of the definitions because our notion of integral (in the presence of decency) is equivalent to reduced and irreducible. $\square$

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