# Show that $O_k$ is a principal ideal domain if and only if

Let $$k$$ be an algebraic number field.cShow that $$O_k$$ is a principal ideal domain if and only if it satisfies the following condition: for every $$\alpha\in k$$, but $$\alpha\not\in O_k$$,there are $$\beta,\gamma\in O_{k}$$ such that $$0<|\alpha\beta-\gamma|<1$$. I want to prove $$h(k)=1$$. But I don't know how to find $$\beta$$$$\gamma$$