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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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