Remark

Tag 0CWN, permanent. Cite it as [Bourbaki, Tag 0CWN].

Commutative Algebra, chapter IX, § 2. Anneaux de Cohen

Suppose that $p1_A$ is not nilpotent (this takes place in particular when $A$ is an integral ring whose field of fractions is of characteristic 0). Then $C$ is a discrete valuation ring whose field of fractions is of characteristic 0.

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

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