In this question [Hilbert class field of Quadratic fields](https://mathoverflow.net/questions/76628/hilbert-class-field-of-quadratic-fields) it is mentioned that if $d\equiv 1 \mod 4$ then the Hilbert class field of $\mathbb{Q}(\sqrt{-d})$ contains $\mathbb{Q}(i,\sqrt{d})$. 


Could someone point me to where the intersections of Hilbert class fields of imaginary quadratic fields is discussed in more detail?



Intersection of Hilbert class fields of imaginary quadratic fields
The generalization of the phenomena you see is *genus theory*. If $K = \mathbf{Q}(\sqrt{-d})$ and $H = K(j\_d)$ then $H$ contains the Genus field $G$.


If $d = \prod\_{i=1}^n p\_i$ is squarefree (and odd for convenience's sake) then $G = K(\sqrt{p\_i^\*})$ where $p\_i^\* = (-1)^{(p\_i-1)/2}p\_i$. In particular $p\_i^\* = p\_i$ if and only if $p\_i \equiv 1\bmod 4$. Therefore if $d \equiv 1\bmod 4$ then $d = \prod\_i p\_i^\*$. Therefore if $d \equiv 1\bmod 4$ then $G$ and thus $H$ contains $K(\sqrt d) = K(i)$.


In general, if $d\_1$ and $d\_2$ have lots of prime factors in common, then the genus fields of their corresponding imaginary quadratic fields will also have large intersections. Outside of the genus field, I don't know of any studies into the intersections of Hilbert Class Fields.


The standard reference for this is Cox's wonderful book "Primes of the form $x^2 + ny^2$."


