Exercise 28, § 4

Let $A$ be an integrally closed noetherian local ring, $K$ its field of fractions, $x$ an element of $K - A$, $B$ the ring $A[x]$ and $p$ a maximal ideal of $B$. Let $a$, $b$ be nonzero elements of $A$ such that $bx = a$. a) The ring $B\left[\frac{1}{b}\right]$ is integrally closed. b) There exists a monic polynomial $f$ with coefficients in $A$ such that $f(x) \in p$. c) Let $E$ be the field obtained by adjoining the roots of $f(X)$ to $K$, $A’$ the integral closure of $A$ in $E$, $B’$ the ring $A’[x]$. Let $p’$ be a maximal ideal of $B’$ lying above $p$. Then the integral closure of $B’_{p’}$ is finite over $B’_{p’}$. (Use the preceding exercise.) d) The integral closure of $B’_p$ is finite over $B’_p$. (Use Exercise 26.) e) The integral closure of $B_p$ is finite over $B_p$.