So I'm trying to show that if we have some number field $k/\mathbb{Q}$ and ring of integers $R_k\subset k$, and an element of $R_k$, say $\alpha$, that the field norm of $\alpha$ is equal to the "norm" of the ideal $(\alpha)$, which from what I can understand is defined to be the number of elements of $R_k/(\alpha)$. I have found a proof of this fact in W. Narkewiecz's "Elementary and Analytic Theory of Algebraic Numbers" (pp. 57-58) which uses a huge number of symbols, defining scads of sums and little niggling elements all over the place apparently arbitrarily (aside from the fact that it all works out in the end). I guess I could spend an hour trying to figure all this out, but this is a sort of homework assignment and I don't really want to just copy this down. Is there any sensible, intuitive way of understanding what is going on here, or is truly just a nice effect of all the symbols coming out the right way? (A better, more elementary reference might be helpful as well...)
Thanks