# Field norm and ideal norm

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• Jul 22nd 2010, 08:38 AM
dmz
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 $N(I)=|N(\alpha)|$?

Thanks in advance.
• Jul 22nd 2010, 03:43 PM
tonio
Quote:

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 $N(I)=|N(\alpha)|$?

Thanks in advance.

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.

Tonio
• Jul 22nd 2010, 09:47 PM
dmz
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...