Equivalent assertions for a quadratic number field

Question

Let $K$ be a quadratic number field with ring of integers $\mathcal O_K$. Why are the following two assertions equivalent?

• $\mathcal O_K$ is factorial
• $\mathcal O_K$ is a principal ideal domain

Thanks for any hint!

Answer 1

Ring $\mathcal{O}_K$ is a Dedekind domain. Now there is the following fact.

A Dedekind domain $A$ is UFD if and only if it is PID.

Let me give you a sketch of proof.

The implication $\Leftarrow$ is clear. The implication $\Rightarrow$ follows from the fact that in UFD every prime ideal of height $1$ is principal. Now in a Dedekind domain by definition every prime ideal is either zero or of height one. Thus if $A$ is both a Dedekind domain and UFD, then every prime ideal in $A$ is principal. Moreover, every ideal in a Dedekind domain can be uniquely decomposed onto a product of prime ideals. Thus every ideal in $A$ can be uniquely decomposed onto a product of principal ideals. Hence every ideal in $A$ is principal, because it is a product of principal ideals.