I want to find all the prime ideals of the ring of integers of bi-quadratic fields. As I know, every prime ideal $\mathcal{P}$ of an algebraic number field $K = \mathbb{Q}(\alpha)$ lies over an ideal generated by a prime $p$ in $\mathbb{Z}$. And, if the prime $p$ in $\mathbb{Z}$ divides $[\mathcal{O}_K : \mathbb{Z}[\alpha]]$ then from Prime ideals of the ring of integers of an algebraic number field I got explicitly all primes $\mathcal{P}$ in $\mathcal{O}_K$ that lies over $p$. Now my first question is how can I find the other prime ideals $\mathcal{P’}$ in $\mathcal{O}_K$ ? \
And my second question is that for these type of prime ideals $\mathcal{P’}$ how the field $\mathcal{O}_{K}/ \mathcal{P’}$ looks like?(as I know, for other cases $\mathcal{O}_{K}/ \mathcal{P}$ is isomorphic to some finite field).
For this, I have to calculate first [$ \mathcal{O}_K : Z[\alpha] $] for any bi-quadratic field $ K = \mathbb{Q}(\alpha) $. How can I do this?