Let $K$ be a CM field and $F$ the maximal totally real subfield of $K$. We assume the ring of integers $\mathcal{O}_K$ is the form $\mathbb{Z}[\alpha, \beta]$. Then can we determine the form of $\mathcal{O}_F$? For example, assume $K= \mathbb{Q}(\zeta)$ be a cyclotomic field, then $\mathcal{O}_K = \mathbb{Z}[\zeta]$ and $\mathcal{O}_F = \mathbb{Z}[\zeta + \zeta^{-1}]$. From this example, I expect that $\mathcal{O}_F = \mathbb{Z}[\alpha + \beta, \alpha\beta]$, but I don't know the proof.
Also, can this fact be generalized a little more?
Thank you.