# How does the theory of the quadratic number fields relate to the quadratic forms?

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.

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For a start see en.wikipedia.org/wiki/Narrow_class_group . This should also be discussed in Cox's Primes of the form $x^2 + ny^2$. Please do not keep commenting on your own question; I don't really see what good it's doing. – Qiaochu Yuan Apr 2 '11 at 16:31
Thanks. Qiaochu Yuan. – awllower Apr 2 '11 at 16:34
See Jay Goldman's "The Queen of Mathematics : An Historically Motivated Guide to Number Theory". It sets out the correspondence between primitive binary quadratic forms with a fundamanetal discriminant and ideals in the ring of integers of a number field. This leaves out the case of quadratic forms with general nonsquare discriminant, which will correspond to ideals classes in a proper subring of the integers of a quadratic field. – KCd Oct 9 '11 at 13:22
Really thanks @KCd !! – awllower Oct 19 '11 at 8:05

For a different take, see Edwards's Higher Arithmetic.

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Oh, thanks. Ihf. – awllower Apr 4 '11 at 2:13

I wrote up an 8 page summary of forms and their relation to ideals at

I would also recommend Lemmermeyer's book `binary quadratic forms' searchable on google.

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Thanks for the summary, from which I will start to benefit, or I so hope. – awllower Oct 19 '11 at 8:03
It seems that the link fails now. – awllower Nov 18 '14 at 13:17