Proof of a property of PID Property: If $F$ is an integral domain and $c$ is an irreducible element in $F$, then $F[x]$ is not a PID(principal ideal domain. 
I saw the property in my textbook, but I wander how to proof it. 

Recall that in a PID each non-zero prime ideal is maximal.
Note that $(X)$ is a prime ideal in $F[X]$ that is not maximal; as $F[X]/(X)$ is isomporphic to $F$ which is a domain (by assumption) yet not a field (as it contains an irreducible element). 
More explicitly, you could argue that $(c,X)$ cannot be generated by a singel element. 
