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Lemma 10.47.9. Let $K/k$ be a geometrically irreducible field extension. Let $S$ be a geometrically irreducible $K$-algebra. Then $S$ is geometrically irreducible over $k$.

Proof. By Definition 10.47.4 and Lemma 10.47.3 it suffices to show that the spectrum of $S \otimes _ k k'$ is irreducible for every finite separable extension $k'/k$. Since $K$ is geometrically irreducible over $k$ we see that $K' = K \otimes _ k k'$ is a finite, separable field extension of $K$. Hence the spectrum of $S \otimes _ k k' = S \otimes _ K K'$ is irreducible as $S$ is assumed geometrically irreducible over $K$. $\square$

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