Exercise 22, § 1

Let $A$ be an integrally closed domain and $K$ its field of fractions; show that for every extension L of K there exists an integrally closed subdomain B of L whose field of fractions is L and such that $B \cap K = A$ (consider L as an algebraic extension of a pure transcendental extension $E = K(X_i)_{i \in I}$ of K and thus reduce it to the case where L is an algebraic extension of K, using Exercise 11 (a)).