# Class numbers in the unramified biquadratic extensions of number fields

Let $$K/k$$ be an unramified biquadratic extension of number fields (i.e., $$\operatorname{Gal}(K/k)\simeq V_4$$), and $$k_1$$, $$k_2$$ and $$k_3$$ its three intermediate fields. I know, in general, we can obtain the class numbers $$h(K)$$ of $$K$$ from the knowledge of $$h(k_1)$$, $$h(k_2)$$ and $$h(k_3)$$ (see the paper of Yamamura Ken: "Maximal unramified extensions of imaginary quadratic number fields of small conductors," Lemma 2, p.418).

My question is, can we get any information about $$h(k_3)$$ by knowing $$h(k_1)$$ and $$h(k_2)$$? For example, if we had a prime $$p$$ that divides $$h(k_1)$$ and $$h(k_2)$$ at the same time, can we say that $$p$$ divides $$h(k_3)$$ too? In general, I know there is no simple relation between them; I just want to know if we can get something when we add the condition of unramification.

2. It is not true, as you write, that the class number of $$K$$ can be determined from the class numbers of the $$k_i$$; you also need information on the behavior of the unit groups.
3. MO is for research: Take primes $$p$$ and $$q$$ and the quadratic number fields $$k_1$$, $$k_2$$ generated by their square roots and see whether you can come up with a pattern for the class number of $$k_3$$. Apart from Scholz's reflection theorem and inequalities regarding the 4-rank of the class groups, nothing is known, and no nontrivial relation is expected to exist.
4. If $$K/k$$ is unramified, then you do get information on $$2$$-class numbers of the subfields as long as the $$2$$-class number of $$K$$ is small. If $$K$$ has odd class number, the $$2$$-class numbers of the $$k_i$$ are all equal to $$2$$. In general, however, you only find that the $$2$$-class numbers of the $$k_i$$ divide $$2 h(K)$$ (class field theory at work).
5. It is rather easy to prove that the odd part of the relative class group of $$K$$ is the direct product of the odd parts of the relative class groups of the $$k_i$$. Apart from this, nothing is known.