Exercise 20, § 2

Let $A$ be the integral domain defined in Chapter III, § 3, Exercise 15 (b), $C$ the local ring $\mathbf{A}_m$, $n = mA_m$ the maximal ideal of $C$ and $L$ the field of fractions of $A$ and $C$. If $x, y$ are the canonical images of $X$ and $Y$ in $C$, show that $t = y/x$ does not belong to $C$ but that $t^2 \in C$, so that $C$ is not integrally closed; moreover $C' = C[t]$ is the integral closure of $C$ and is isomorphic to $S^{-1}K[T]$ (T an indeterminate), where $S$ is the multiplicative subset of $K[T]$ consisting of the polynomials which are not divisible by $T - 1$ nor by $T + 1$. Show that $C'$ is a finitely generated $C$-module and that the (maximal) ideals of $C'$ lying above $n$ are the principal ideals $q_1 = (t - 1)C'$ and $q_2 = (t + 1)C'$; but the local rings $C'_{q_1}$ and $C'_{q_2}$ are not $C$-algebras integral over $C$.