= Solution
By <unique factorization of ideals in a number field>, each nonzero integral ideal has a unique expression
$$
\mathfrak a=\prod_{\mathfrak p}\mathfrak p^{e_{\mathfrak p}},
$$
with only finitely many nonzero exponents. Since the <ideal norm> is multiplicative, formally expanding one geometric series for each prime ideal gives
$$
\begin{aligned}
\prod_{\mathfrak p}
\left(1-N(\mathfrak p)^{-s}\right)^{-1}
&=\prod_{\mathfrak p}\sum_{e\geq0}N(\mathfrak p)^{-es}\\
&=\sum_{0\ne\mathfrak a\subseteq\mathcal O_K}N(\mathfrak a)^{-s}\\
&=\zeta_K(s).
\end{aligned}
$$
Uniqueness of ideal factorization says that every term occurs exactly once. This proves the <Euler product for the Dedekind zeta function> as a formal identity.
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