I want to understand how to generalize standard facts in number theory on rational integer ideals to more general number fields. We know that for instance $$\# \{k \in \mathbb{Z} \ : \ 1 \leq k \leq N\} = N.$$
For a number field $F$ with ring of integers $\mathfrak{o}$, what can be said of $$\#\{\mathfrak{n} \subset \mathfrak{o} \ : \ 0 \leqslant \text{norm}(\mathfrak{n}) \leqslant N\} \quad ?$$
and is it straightforward from the first result?
One has the general asymptotic
$$\#\{\mathfrak{n} \subset \mathfrak{o} \ : \ 0 \leqslant \text{norm}(\mathfrak{n}) \leqslant N\} = \rho N + O(N^{1 - 1/n}) $$
where $ n = [F : \mathbf Q] $ and the quantity $ \rho > 0 $ depends on the number field $ F $. (This lemma is used in the proof of the analytic class number formula.) You can find a proof (along with the full statement of the result, with the actual expression for $ \rho $) here.