Exercise 15, § 2

Let $A$ be a ring and $p$ a prime ideal of $A$. Show that, in the polynomial ring $B = A[X]$, the ideal $pB = q$ is prime; the maximal ideal of the local ring $B_q$ is equal to $pB_q$; if $k$ is the field of fractions of $A/p$, $B_q/pB_q^l$ is isomorphic to the field of rational fractions $k(X)$. If $A$ is integrally closed, so is $B$.