Corollary 1

Let $A$ be an integral domain and $A'$ its integral closure. Then the integral closure of the polynomial ring $A[X_1, \ldots, X_n]$ is $A'[X_1, \ldots, X_n]$.

Arguing by induction on $n$, the problem is immediately reduced to the case $n = 1$. Let $K$ be the field of fractions of $A$, which is also that of $A'$; if an element $P$ of the field of fractions $K(X)$ of $A[X]$ is integral over $A[X]$, it belongs to the polynomial ring $K[X]$, for the latter is a principal ideal domain (Algebra, Chapter IV, § 1, no. 5, Proposition 7) and hence integrally closed (Proposition 10); the corollary then follows from Proposition 13 applied to $\mathbf{R} = K$.