Corollary

Tag 04S5, permanent. Cite it as [Bourbaki, Tag 04S5].

Commutative Algebra, chapter VI, § 1. Valuation rings

Every local subring $A$ of a field $K$ is dominated by at least one valuation ring of $K$ .

Apply Theorem 2 to the canonical homomorphism $h$ from $A$ to an algebraic closure $L$ of $A/m(A)$ .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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