Exercise 11, § 4

Let A be a noetherian integral ring, K its field of fractions, E a finite extension of K, $\overline{A}$ the integral closure of A in E. a) For every prime ideal $p$ of A, there are only a finite number of prime ideals $\mathfrak{P}$ of $\overline{A}$ lying over $p$, and the fields $\kappa(\mathfrak{P})$ are finite extensions of $\kappa(p)$. (Reduce to the case where A is a local ring with maximal ideal $p$ and where $E = K$, and use Exercise 10.) b) Let P be the set of prime ideals of height 1 of $\overline{A}$. If $p \in P$, then $\overline{A}_p$ is a discrete valuation ring. c) We have $\overline{A} = \bigcap_{p \in P} \overline{A}_p$ (in E).