Exercise 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 $p' = n \cap A'$. Then, if the local rings $A'_p$ and $B_{n'}$ are distinct, there exists a prime ideal of $B$ contained in $n$, distinct from $n$ and containing $p$ ("Zariski's Principal Theorem"). (Using Exercise 6 (b), reduce it to the case where $A' = A$, then reduce it to the case where $A$ is a local ring with maximal ideal $p$. Show that $B/n$ is then a finite algebraic extension of $A/p$ (cf. no. 4, Theorem 3); deduce that, for every ring $C$ such that $A \subset C \subset B$, $n \cap C$ is a maximal ideal of $C$. 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).)