Let $K$ be a number field and $\mathcal{O}_K$ be the ring of integers of $K$. For a non-zero ideal $\mathfrak{a}$ in $\mathcal{O}_K$, the norm $N(\mathfrak{a})$ is defined as the cardinality of quotient ring $\mathcal{O}_K / \mathfrak{a}$ , that is $N(\mathfrak{a}) = |\mathcal{O}_K / \mathfrak{a}|$.
I showed that $N(\mathfrak{p}^n) = N(\mathfrak{p})^n$ for all $n \geq 1$. I also know that $\mathcal{O}_K$ is a Dedekind domain. Thus, every non-zero ideal can be written uniquely as a product of prime ideals.
Now, I want to prove that norm is a completely multiplicative function. That is, $$ N(\mathfrak{a}\mathfrak{b}) = N(\mathfrak{a})N(\mathfrak{b}) $$ for any ideals $\mathfrak{a}, \mathfrak{b}$ in $\mathcal{O}_K$.