In my professor's algebra lecture notes she says that in a unique factorization domain (UFD) of the form $\mathbb Z[\sqrt D]$ (where $D$ is some integer) the prime elements are those that have norm $p$ or $p^2$ for some prime $p \in \mathbb Z$.
Say, $P$ is a prime element in such a UFD. Then if $P \mid A B$ it implies $P \mid A$ or $P \mid B$. This means $\mathcal N(P) \mid \mathcal N(A) \mathcal N(B) \implies \mathcal N(P) \mid \mathcal N(A)$ or $\mathcal N(P)|\mathcal N(B)$.
I understand that if $\mathcal N(P)$ is a prime number then $\mathcal N(P) \mid \mathcal N(A) \mathcal N(B) \implies \mathcal N(P) \mid \mathcal N(A)$ or $\mathcal N(P)|\mathcal N(B)$ must hold true. But why must it also hold true if $\mathcal N(P) = p^2$ for some prime in $\mathbb Z$. Secondly, why only $p$ and $p^2$? Why can't $\mathcal N(P)$ be something else?