Lemma 15.108.2. Let $(A, \mathfrak m)$ be a Noetherian local ring. The number of branches of $A$ is the same as the number of branches of $A^\wedge$ if and only if $\sqrt{\mathfrak qA^\wedge }$ is prime for every minimal prime $\mathfrak q \subset A^ h$ of the henselization.

Proof. Follows from Lemma 15.108.1 and the fact that there are only a finite number of branches for both $A$ and $A^\wedge$ by Algebra, Lemma 10.31.6 and the fact that $A^ h$ and $A^\wedge$ are Noetherian (Lemma 15.45.3). $\square$

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