Exercise 14, § 1

Let A be a completely integrally closed domain and K its field of fractions. Show that for every extension L of K the integral closure A of A in

L is a completely integrally closed domain. (Reduce it to the case where L is a finite quasi-Galois extension of K of degree m: if x ∈ L is such that there exists d ∈ A' such that dx^n ∈ A' for all n ≥ 0, show first that it may be assumed that d ∈ A, then deduce that, if c_i (1 ≤ i ≤ m) are the coefficients of the minimal polynomial of x over K, then d^m c_i^n ∈ A for all n ≥ 0.)