$\mathcal{O}_K$ UFD $\iff h_K=1$

Question

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?

Answer 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.

Answer 2

For Dedekind rings PID and UFD is equivalent. Since the ring of integers $\mathcal{O}_K$ of a number field $K$ is a Dedekind ring, it follows that UFD is the same as PID for it, i.e., the same as class number $1$.