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 that ?

Thanks in advance.

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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 that ?

Thanks in advance.

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For this you need the machinery from algebraic number theory: as is the cardinality of the ring and this number is, if is a principal ideal, the norm of , then you get the result.

Tonio

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...