Recently, I became interested in the idea of the class number of a number field $K$.
The class group of a number field $K$ (or of $O_K$, the ring of integers of $K$) is the quotient group $Cl_K$ (or $Cl_{O_K}$ or $Cl(K)$) given by $Cl_K = $ (fractional ideals of $O_K$)/(principal fractional ideals of $O_K$). The order of the class group is called the class number of $K$. The class number of a number field $K$ is always finite.
The class group, then, serves as a measure of the failure of $O_K$ to be a principal ideal domain (and hence, a unique factorization domain, since $O_K$ is a Dedekind domain). For example, $K$ has trivial class group $\iff$ $O_K$ is a unique factorization domain. If one is handed a quadratic field $\mathbb{Q}(\sqrt{d})$ ($d \neq 0,1$ is a square-free integer), one can calculate its ring of integers, and determine its class number from there.
From what I understand, Gauss developed tools involving quadratic forms to compute class numbers of quadratic fields, and this theory has a slightly different flavor for real quadratic fields compared to imaginary quadratic fields.
I would like a easy to understand reference for Gauss' work involving quadratic forms to compute the class number of quadratic fields. I don't have much background in number theory, for example, and I'm primarily a commutative algebra student. It would be nice to see how Gauss' methods compare to the conventional methods of computing class groups of quadratic fields and observing their orders to determine the class number of that number field.
Some looming questions, as well. Did Gauss' theory improve the theory of class groups ? Or was it Gauss' theory that introduced class groups in the first place ? Is it any less tedious to use Gauss' theory than to just explicity compute the class group of a quadratic field ? When people refer to Gauss' theory of quadratic forms to compute class numbers of quadratic fields, does this include discriminants and the Minkowski bound ? Or are they strictly referring to Gauss' "class number problem" ?
Thank you for all of your time in help. Mathematics in quarantine is the best. (=