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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?

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Question: "I got explicitly all primes P in OK that lies over p. Now my first question is how can I find the other prime ideals P′ in OK? And my second question is that for these type of prime ideals P′ how the field OK/P′ looks like?(as I know, for other cases OK/P is isomorphic to some finite field)."

Answer: If $K$ is any number field and $A$ is the ring of integers in $K$, the inclusion map $\mathbb{Z} \subseteq A$ is integral. Any non-zero prime ideal $\mathfrak{m} \subseteq A$ is maximal and $\mathfrak{m} \cap \mathbb{Z}=(p)$ for some prime number $p$. The field extension

$$F1.\text{ }\mathbb{F}_p \subseteq A/\mathfrak{m}$$

is finite hence there is an integer $r \geq 1$ with $A/\mathfrak{m} \cong \mathbb{F}_{p^r}$. Note that since $(p):=\mathfrak{m} \cap \mathbb{Z}$ it follows there is an inclusion $(p) \subseteq \mathfrak{m}$ and a well defined injective map

$$\phi: \mathbb{F}_p \rightarrow A/\mathfrak{m}$$

Hence any prime in $A$ "lies over" a prime ideal in $\mathbb{Z}$ and the isomorphism $F1$ holds for all maximal ideals $\mathfrak{m} \subseteq A$ (the prime number $p$ depends on the ideal $\mathfrak{m}$).

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  • $\begingroup$ Do you want to say that $\mathcal{O}_K / \mathcal{P’}$ is a finite field? I don’t think so. I am not able to give you counter example now. But , it should be an infinite field. $\endgroup$ Jun 17, 2021 at 12:49
  • $\begingroup$ @SrijoneeShabnam - if $K$ is a number field with ring of integers $A$, it follows for any maximal ideal $m \subseteq A$ with $m\cap \mathbb{Z}=(p)$, the extension $\mathbb{Z}/p\mathbb{Z} \subseteq A/m$ is finite. $\endgroup$
    – hm2020
    Jun 17, 2021 at 14:09
  • $\begingroup$ How can you claim that $\mathbb{Z}/p \mathbb{Z} \subseteq A/m $ ? It is not clear to me. $\endgroup$ Jun 17, 2021 at 14:16

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