Proposition 1

Let A be a noetherian integral ring, K its field of fractions. Suppose that for every finite radicial extension L of K, the integral closure of A in L is a finite A-algebra. Then the ring A is Japanese.

Let E be a finite extension of K. Let N be a finite quasi-Galois extension of K containing E (A, V, p. 54, cor. 1), and L the field of invariants of the group of K-automorphisms of N. Then (A, V, p. 73, prop. 13), L is a radicial extension of K and N is a separable extension of L. The integral closure B of A in L is therefore by hypothesis a finite A-algebra; the integral closure C of B in N is a finite B-algebra (V, § 1, No. 6, cor. 1 to prop. 18), hence a finite A-algebra. The integral closure of A in E is contained in C, and is therefore a finite A-algebra since A is noetherian.