Exercise 1, § 4

Let $A$ be a semi-local noetherian integral ring of dimension 1. a) The integral closure of $A$ is a finite $A$-algebra if and only if the completion $\hat{A}$ of $A$ is reduced. (If $\hat{A}$ is reduced, use the corollary to Th. 2 of § 4, No. 2. For the implication in the other direction, reduce to the case where $A$ is integrally closed, and argue as in the proof of Th. 3 of § 4, No. 4). b) The following three conditions are equivalent: (i) $A$ is a Japanese ring; (ii) $A$ is a Nagata ring; (iii) $\hat{A}$ is reduced and if $q_1, ..., q_n$ are the minimal prime ideals of $\hat{A}$, then the fields $\kappa(q_i)$ are separable extensions of the field of fractions of $A$.