Exercise 5, § 2

Let $A$ be an integrally closed ring with field of fractions $K$. Given a prime ideal $p$ of $A$, and an element $x$ of $K$ such that $x \notin A_p$, and $x^{-1} \notin A_p$, the ideal $pA[x]$ of $A[x]$ is prime; one has: $pA[x] \cap A = p$ and the canonical homomorphism $(A/p)[X] \to A[x]/pA[x]$ is an isomorphism.