Theorem 2 (Nagata)

Every complete local noetherian integral domain A is Japanese.

By Theorem 3 of § 2, No. 5 and Theorem 2 of § 3, No. 3, there exist an integer $n \geq 0$, a ring R which is a field or a discrete valuation ring with field of fractions of characteristic 0, and a subring B of A, isomorphic to R[[T₁, ..., Tₙ]] and such that A is a finite B-algebra. Then R is Japanese (No. 1, example and corollary to prop. 1), hence B is Japanese (corollary to Theorem 1), and A is Japanese (No. 1, prop. 2).