$h\_{d\_1}, h\_{d\_2}$ and $h\_{d\_1d\_2}$ are class number of $\Bbb Q(\sqrt{d\_1}),\Bbb Q(\sqrt{d\_2}),\ and \ \Bbb Q(\sqrt{d\_1d\_2})$ respectively.



Is there any relation between $h_{d_1},h_{d_2}$ and $h_{d_1d_2}$?
There is a relation between the class numbers of those quadratic fields *and* the class number of the biquadratic field $\mathbf Q(\sqrt{d\_1},\sqrt{d\_2})$. The key term to look for is "Brauer relation", which has a wider scope than just this example. A reference for this topic has already been mentioned in an earlier MO question [here](https://mathoverflow.net/questions/262735/class-number-of-biquadratic-fields?rq=1). See also equations (2.1) and (2.2) of Bosma and de Smit's "Class number relations from a computational point of view" (J. Symbolic Computation ${\mathbf{31}}$ (2001), 97-112).


