Exercise 21, § 4

a) Let $A$ be an integral domain, $K$ its field of fractions. If $K$ is of nonzero characteristic $p$ and if $E$ is a finite radicial extension of the field of rational fractions $K(X)$, there exists a finite radicial extension $F$ of $K$ and a power $q$ of $p$ such that $F(X^{1/q})$ is an extension of $E$. b) Let $A$ be an integrally closed Nagata ring. Then the polynomial ring $A[X]$ is Japanese. c) Let $A$ be an integrally closed Nagata ring. Then every polynomial ring $A[X_1, ..., X_n]$ in a finite number of indeterminates is a Nagata ring.