Exercise 19, § 1

(a) Let A be an integrally closed domain, K its field of fractions, L a separable algebraic extension of K, x an element of L integral over A, K' the field K(x) ⊂ L and F the minimal polynomial of x over K. If x is of degree n over K, show that the integral closure of A in K' is contained in the A-module generated by the elements 1/F'(x), x/F'(x), ..., x^{n-1}/F'(x) of K'. (If z ∈ K' is integral over A, write

zF'(x) = g(x), where g(X) = b_0 + b_1X + ... + b_{n-1}X^{n-1} ∈ K[X].

Show with the aid of Lagrange's interpolation formula that

(b) Let $A$ be an integrally closed domain, $K$ its field of fractions and $p$ the characteristic exponent of $K$. Suppose that the subring $A^{1/p}$ of $K^{1/p}$ consisting of the $p$-th roots of the elements of $A$ is a finitely generated $A$-module. Show that for every finite extension $E$ of $K$ the integral closure of $A$ in $E$ is a finitely generated $A$-module (reduce to the case where $E$ is a quasi-Galois extension).