习题 7, § 3

Let $A$ be an integrally Noetherian domain (Exercise 6), $(t_i)_{1 \leq i \leq n}$ a finite sequence of elements of an extension field of the field of fractions of $A$, $B$ the ring $A[t_1, \ldots, t_n]$, $n$ a maximal ideal of $B$, $p = A \cap n$, $A'$ the integral closure of $A$ in $B$ and $p' = n \cap A'$. 那么，如果局部环 $A'_p$ 和 $B_{n'}$ 不同，则$n$ 中存在$B$ 的素理想，与 $n$ 和包含 $p$ 不同（“Zariski 的校长定理”）。 (利用习题 6 (b), 约化到$A' = A$的情况，然后约化到$A$是局部环，最大理想为$p$。 证明 $B/n$ 是 $A/p$ 的有限代数扩张 (cf. no. 4, 定理 3)；推导出，对于每个环$C$，使得$A \subset C \subset B$、$n \cap C$是$C$的最大理想。 Then argue by induction on $n$, denoting by $B_i$ the integral closure of $A[t_1, \ldots, t_i]$ in $B$; distinguish two cases according to whether $t_{i+1}$ is transcendental or algebraic over the field of fractions of $B_i$; in the former case, use Exercise 4 and, in the latter, Exercise 5 (c).)