Suppose can be generated by a single element. That means that there exist such that and but since so it is impossible.
similarly you can easily prove a general result which is quite useful: if is a non-zero ring, then the polynomial ring is a PID if and only if is a field.
every non-zero element of is a unit, which means is a field. so let let since is a PID, there exists such that hence must
be a unit element of because so i.e. there exists such that now put to get i.e. is a unit element of