# All prime ideals in ring of integers of a bi-quadratic field

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?

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 $$\mathfrak{m}$$).

• 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. Jun 17, 2021 at 12:49
• @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. Jun 17, 2021 at 14:09
• How can you claim that $\mathbb{Z}/p \mathbb{Z} \subseteq A/m$ ? It is not clear to me. Jun 17, 2021 at 14:16