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Math Help - Field norm and ideal norm

  1. #1
    dmz
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    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.
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  2. #2
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    Quote Originally Posted by dmz View Post
    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
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  3. #3
    dmz
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    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...
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