习题 20, § 2

设$A$为章III, § 3、习题15(b)中定义的整环、$C$局部环$\mathbf{A}_m$、$n = mA_m$$C$和$L$的最大理想$A$ 分数域和 $C$。 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 $T + 1$. 证明$C'$是有限生成的$C$模，并且$C'$位于$n$之上的（极大）理想是主理想$q_1 = (t - 1)C'$和$q_2 = (t + 1)C'$；但局部环 $C'_{q_1}$ 和 $C'_{q_2}$ 不是 $C$ 在 $C$ 上的代数积分。