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

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

Thanks in advance.

Printable View

- Jul 22nd 2010, 07:38 AMdmzField 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 $\displaystyle I=\alpha O_K$ that $\displaystyle N(I)=|N(\alpha)|$?

Thanks in advance. - Jul 22nd 2010, 02:43 PMtonio

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

Tonio - Jul 22nd 2010, 08:47 PMdmz
Yes. I think the result must follow from the definition of field norm, but it's not obvious for me, how.

So, seeking for machinery...