$\mathcal{O}_K$ UFD $\iff h_K=1$ How can we prove that, if $K$ is a number field, then his integer ring $\mathcal{O}_K$ is an unique factorization domain if and only if the class number of $K$ is 1?

Consider these:


*

*The ideals of $\mathcal{O}_K$ have unique factorization into prime ideals.

*The class number of $K$ is $1$ iff every ideal of $\mathcal{O}_K$ is principal.
