Fact about sum of ideal classes in the ideal class group

Question

Let $K$ be a number field with ring of integers $\cal{O}_K$ and class group $\cal{O}_K$. Let $\mathfrak{p},\mathfrak{q}$ be two integral ideals such that $\mathfrak{p}\mathfrak{q} = (\alpha)$ for some $\alpha \in \cal{O}_K$. Then, why do we have that $[\mathfrak{p}]+[\mathfrak{q}] = \bar{0}$ in $Cl(\cal{O}_K)$?

Answer 1

The author of the answer you linked in the comments refers to the ideal class of the principal ideal $\mathfrak{p}\mathfrak{q} = (\alpha)$ for some $\alpha \in \mathcal{O}_K$. In $\operatorname{Cl}(\mathcal{O}_K)$, this means that $[\mathfrak{p}\mathfrak{q}] = \bar{1}$, or that $[\mathfrak{p}][\mathfrak{q}] = \bar{1}$ (when we write everything multiplicatively). If we write additively (which is often done to highlight the commutativity of a group), we have $[\mathfrak{p}] + [\mathfrak{q}] = \bar{0}$. Note that in the answer you linked, the author goes on to show that the class group is isomorphic to $C_2 \times C_2$, which is an abelian group, so the additive notation highlights this.