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.
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.
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