The Stacks project

Lemma 10.50.7. Let $L/K$ be an extension of fields. If $B \subset L$ is a valuation ring, then $A = K \cap B$ is a valuation ring.

Proof. We can replace $L$ by the fraction field $F$ of $B$ and $K$ by $K \cap F$. Then the lemma follows from a combination of Lemmas 10.50.4 and 10.50.5. $\square$

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