Relating binary quadratic forms and quadratic number fields

Question

In the context of integral binary quadratic forms, there is the concept of the class group $\mathcal{C}(d)$ of a discriminant $d$. And in the context of number fields, there is the concept of the ideal class group $Cl_K$ of a number field $K$. About these I have a few questions:

1. Is it true that for a quadratic number field $K$, that $Cl_K\cong\mathcal{C}(d)$, where $d$ is the discriminant of $K$?
2. Where can I find a reference of this?
3. Can this be generalized to cubic number fields, quartic number fields, ... etc.?
4. Thus it seems that only a portion of the theory of integral binary quadratic forms (those with discriminant fundamental) correspond with the theory of quadratic number fields. Why is this the case?