As every one knows, the quadratic number fields shares a deep connection with the binary quadratic forms; I have been told this relation when I was a senior high, and now I, learning some difficult theories like the theory of fields of classes, know some basic facts about the algebraic number fields of degree two but I know nothing about what is going on in the Disquisitiones Arithmeticae by Gauss even though I have read it twice; it seems to be a kind of a magic there; he makes things like ideals happen with only elementary and lengthy processes. My question is the
Is it necessary to verify all identities in the book by Gauss to understand his theory of quadratic forms?
I suppose it is not; then what could be said about the connection between them so that one can easily have a sight on what he is doing by modern algebraic tools?
If it was possible, please say something assertive like "The forms are just ..." Thank you very much.