Lemma 33.8.6. Let $X$ be a geometrically irreducible scheme over the field $k$. Let $\xi \in X$ be its generic point. Then $\kappa (\xi )$ is geometrically irreducible over $k$.

**Proof.**
Combining Lemma 33.8.5 and Algebra, Lemma 10.46.6 we see that $\mathcal{O}_{X, \xi }$ is geometrically irreducible over $k$. Since $\mathcal{O}_{X, \xi } \to \kappa (\xi )$ is a surjection with locally nilpotent kernel (see Algebra, Lemma 10.24.1) it follows that $\kappa (\xi )$ is geometrically irreducible, see Algebra, Lemma 10.45.7.
$\square$

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