A prime ideal of a polynomial ring over a PID can be generated by two elements. [duplicate]
Solution 1:
If $P$ contains a non-zero element of $D$, then it contains a prime element of $D$ (why?), say $p$. Then $P/pD[x]$ is a prime ideal of $D[x]/pD[x]$. But $D[x]/pD[x]$ is isomorphic to $(D/pD)[x]$ (why?) and $D/pD$ is a field, so $D[x]/pD[x]$ is a PID. This shows that $P/pD[x]$ is principal. It follows that $P$ can be generated by two elements.