Let $K\_1=\Bbb Q(\sqrt{d\_1})$ , $K\_2=\Bbb Q(\sqrt{d\_2})$ and $K=\Bbb Q(\sqrt{d\_1},\sqrt{d\_2})$.Suppose $h\_1,h\_2,h$ be class number of $K\_1,K\_2,K$ respectively.


(i) Can we express $h$ in terms of $h\_1,h\_2$?


(ii) Knowing the divisibility properties of $h\_1,h\_2$, I want help with concluding about the divisibility of $h\_1,h\_2.$



How is class of composition of two quadratic fields is related class numbers of quadratic field?
I assume you mean $K\_1=\mathbb Q(\sqrt{d\_1})$.


1) If by $K$ you mean $\mathbb Q(\sqrt{d\_1d\_2})$, then there is no simple relation
between $h$ and $h\_1$ and $h\_2$.


2) If by $K$ you mean the quartic biquadratic field $\mathbb Q(\sqrt{d\_1},\sqrt{d\_2})$, a theorem of Herglotz says that $h=h\_1h\_2h\_3/2^j$, where
$h\_3$ is the class number of $\mathbb Q(\sqrt{d\_1d\_2})$ and $j=0,1,2$ which can
be computed in terms of the units of $K$.


