Exercise 4, § 3

Let B be an integral domain and A a subring of B. Suppose that there exist $n \geq 1$ elements $t_1, \ldots, t_n$ of B which are algebraically independent over A and such that B is integral over $A[t_1, \ldots, t_n]$. Let $n$ be a maximal ideal of B and $p = n \cap A$. Then there is a prime ideal q of B contained in $n$, distinct from $n$ and containing $p$. (Reduce it first to the case where A is integrally closed by considering the integral closure A' of A (contained in the field of fractions of B) and the ring $B' = B[A']$; in the case where A is integrally closed, use Theorem 3 of § 2, no. 4.)