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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.

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  1. Kuroda's class number formula does not require the extension to be unramified.
  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.
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  • $\begingroup$ I sincerely appreciate your reply! I was actually reading your paper on Kuroda’s class number formula, and i assumed there is no nontrivial relation even if we add the condition of unramification. I was just waiting for the confirmation! $\endgroup$ Feb 27 at 0:10

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