Suppose $K=\mathbf{Q}(\sqrt{d})$; it's well known that $\mathcal{O}_K$ is $\mathbf{Z}+\frac{D+\sqrt{D}}{2}\mathbf{Z}$, where $D$ is the discriminant.

What is the analogue of this for a CM extension $F(\sqrt{-\alpha})/F$, where $F$ is totally real (of class number one) and $\alpha \in \mathcal{O}_F$ is totally positive?

Is there a canonical element $\xi$ of $F(\sqrt{-\alpha})$ such that $\mathcal{O}_{F(\sqrt{-\alpha})}=\mathcal{O}_F+\xi \mathcal{O}_F$?

I have looked in the literature, but all I can find are various theorems guaranteeing the *nonexistence* of relative integral bases in various situations. However a theorem in Chapter 7 of Narkiewicz's book guarantees that some $\xi$ does exist in the above situation.