Lemma 10.47.11. Let $K/L/M$ be a tower of fields with $L/M$ geometrically irreducible. Let $x \in K$ be transcendental over $L$. Then $L(x)/M(x)$ is geometrically irreducible.

Proof. This follows from Lemma 10.47.10 because the fields $L(x)$ and $M(x)$ are purely transcendental extensions of $L$ and $M$. $\square$

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