Lemma 10.50.8 (0AAV)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.50: Valuation rings
Lemma 10.50.8 (cite)

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Lemma 10.50.8. Let $L/K$ be an algebraic extension of fields. If $B \subset L$ is a valuation ring with fraction field $L$ and not a field, then $A = K \cap B$ is a valuation ring and not a field.

Proof. By Lemma 10.50.7 the ring $A$ is a valuation ring. If $A$ is a field, then $A = K$ . Then $A = K \subset B$ is an integral extension, hence there are no proper inclusions among the primes of $B$ (Lemma 10.36.20). This contradicts the assumption that $B$ is a local domain and not a field. $\square$

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