I have to prove that is a unique factorization domain that is not a principal ideal domain.
I´m supposed to notice that cannot be generated by an ideal of one element.
Can anyone explain me why is that?
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