Exercise 10, § 1

Let A be a completely integrally closed domain. Show that for every set I the polynomial domain $A[X_i]_{i \in I}$ and the domain of formal power series $A[[X_i]]_{i \in I}$ (Algebra, Chapter IV, § 5, Exercise 1) are completely integrally closed. (Use Proposition 4 of no. 4 and the following lemma: for every finite subset J of I and all $P \in A[[X_i]]_{i \in J}$, let $P_J$ be the formal power series in $A[[X_i]]_{i \in J}$ derived from P by replacing by 0 in P all the X, of index $i \notin J$; if P, Q are two formal power series such that $P_J$ is divisible by $Q_J$ in $A[[X_i]]_{i \in J}$ for all J, then P is divisible by Q in $A[[X_i]]_{i \in I}$. For this note that, if J, J' are two finite subsets of I such that $J \subset J'$ and $P_{J'} = L' Q_{J'}$, and $P_{} = L Q_J$, then L is obtained from L' by replacing by 0 in L' the X, of index $i \in J' - J$.)