Number of ideals with norm equals to a certain number in number field.

Question

I can't follow the claim in the lecture notes below about the number of ideals with norm $n$. Any help or hint is appreciated.

Answer 1

You meant $n^k$. If $N(I)=p^k$ then $I$ is a product of at most $k$ prime ideals of norm a power of $p$ and there are at most $n$ of those.

If $p$ splits completely then the exact number is ${n+k-1 \choose k}$ (a combination with repetition).