Remark 5

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

Commutative Algebra, chapter VIII, § 7. Multiplicités

It may happen that $e(A) = 1$ without A being regular (p. 104, exerc. 5). In fact, a local noetherian ring A is regular if and only if $\hat{A}$ is integral and if one has $e(A) = 1$ (p. 108, exerc. 24).

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

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