Math Help - Field norm and ideal norm

1. Field norm and ideal norm

Hi, could someone please explain the last part from Field norm - Wikipedia, the free encyclopedia:

Why it is so expected for principal ideal $I=\alpha O_K$ that $N(I)=|N(\alpha)|$?

2. Originally Posted by dmz
Hi, could someone please explain the last part from Field norm - Wikipedia, the free encyclopedia:

Why it is so expected for principal ideal $I=\alpha O_K$ that $N(I)=|N(\alpha)|$?

For this you need the machinery from algebraic number theory: as $N(I)$ is the cardinality of the ring $O_k$ and this number is, if $I= \alpha O_k$ is a principal ideal, the norm of $\alpha$ , then you get the result.