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Given a quadratic form with discriminant $D$, what does the class number of $\mathbb{Q}(\sqrt{D})$ tell us?

(This question is inspired by a comment on the question here)

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    Lisa, what else do you want to know?2014-09-23
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    Maybe she should have gone ahead and posted her question even though the system was dead set on finding a "duplicate."2014-09-24
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    @WillJagy I know that $\mathcal{O}_{\mathbb{Q}(\sqrt{D})}$ has unique factorization if its class number is 1, and doesn't if the class number is 2 or higher. But what else does the class number us? Let's say $\mathcal{O}_{\mathbb{Q}(\sqrt{m})}$ has class number 2 and $\mathcal{O}_{\mathbb{Q}(\sqrt{n})}$ has class number 3. Neither is a UFD. Obviously $m \neq n$. Without knowing the actual values of $m$ and $n$, can we contrast $\mathcal{O}_{\mathbb{Q}(\sqrt{m})}$ and $\mathcal{O}_{\mathbb{Q}(\sqrt{n})}$ in any meaningful way?2014-09-24
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    @Lisa, i suggest you work up a few examples where the contrast you want can be found, with a few examples where the distinction you want is not forthcoming, then post a separate question with the examples. there are people on MSE who can answer any related question as long as it is clear what the question might be. It also helps if you get across your current background. Anonymity is all well and good, but not when it hides what level your answers need to be.2014-09-24
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    @Will Well, I'm an amateur. I did reasonably well in high school algebra. As for anonymity, I don't feel like talking about male privilege right now.2014-09-24
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    @Lisa, right. I meant to include your background in some manner when asking a question, even while keeping your identity to yourself; it helps. If you have studied quite a bit on your own, indicate that somehow, particular book titles for example.2014-09-24
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    @Will The ring with $m$ has numbers with multiple factorizations but in each case the total number of factors is the same, but the one with $n$ there can be multiple factorizations of different lengths. e.g., $\mathbb{Z}[\sqrt{-14}]$ has class number 3, and for 81 we have $3^4 = (5 - 2 \sqrt{-14})(5 + 2 \sqrt{-14})$. So, as Finch says, "the class number measures how far $\mathcal{O}_D$ is from being a UFD."2014-09-25
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    @RobertSoupe, I imagine that is the sort of thing Lisa wants. Perhaps you could put an answer with information about the reference Finch, quotes and so forth2014-09-25
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    http://www.people.fas.harvard.edu/~sfinch/csolve/clss.pdf by http://www.people.fas.harvard.edu/~sfinch/2014-09-25
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    @WillJagy The Harold Finch paper is available on his Harvard page: http://www.people.fas.harvard.edu/~sfinch/csolve/clss.pdf But the tricky part here is that the original asker said "binary quadratic forms" but the bounty offerer is talking about rings.2014-09-25
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    @RobertSoupe, this guy is Steven. Harold Finch is a character on http://en.wikipedia.org/wiki/Person_of_Interest_%28TV_series%292014-09-25
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    @WillJagy Oops, I meant to cite Harold Whistler's dissertation "Ethical Considerations of High Frequency Econometrics." ;-)2014-09-25

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I guess you mean a binary quadratic form, i.e. of the form $a x^2 + b x y + c y^2$, with $a,b,c$ integers and of discrmimant $D$ (i.e. $b^2 - 4 a c = D$).

If $\mathbb Q(\sqrt{D})$ has class number one, then the conditions for solving $a x^2 + b x y + c y^2 = p$ ($p$ a prime not dividing $D$) depend only on the residue class of $p$ mod $D$, in fact only on the value of the Kronecker symbol of $p$ mod $D$.

If the class group is just a product of cyclic groups of order 2 (note that this can't be detected by the class number alone, which e.g. can't distinguish between $C_2 \times C_2$ and $C_4$) then the condition for solving $a x^2 + b x y + c y^2 = p$ depends only on the residue class of $p$ mod $D$, but one has to consider not just the Kronecker symbol mod $D$, but other Kronecker symbols modulo various divisors of $D$.

If the class group is not a product of cyclic groups of order 2 (e.g. if the class number is not a power of two) then there is no congruence condition on $p$ which guarantees being able to solve $a x^2 + b x y + c y^2$.

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A geometric interpretation of the class number of any number field is described at http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/SL2classno.pdf.

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I will put a little something here, Lisa wants to know something else but I am not yet sure what that is.

The question linked to (long since deleted) had to do with the fact that the binary quadratic form $$ f(x,y) = x^2 + x y + 3 y^2 $$ is the only form ($SL_2 \mathbb Z$ equivalence class) of its discriminant, $-11.$ This has any number of consequences. For example, if $p, q$ are distinct primes and we have $$ pq = x^2 + xy + 3 y^3, $$ then we know that we can write both $$ p = s^2 + st+ 3 t^2 $$ and $$ q = u^2 + u v + 3 v^2, $$ all in integers.

For comparison, with $g(x,y) = x^2 + x y + 6 y^2,$ we have $$ g(-1,8) = 377 = 13 \cdot 29. $$ But these two primes are represented by $2 x^2 \pm xy + 3 y^2,$ the other two classes of that discriminant. With $h(x,y) = 2 x^2 + xy+ 3 y^2,$ we get $h(2,1) = 13,$ then $h(-2,3) = 29.$ At most one class (along with its opposite) represents a prime.

I also put some stuff at http://math.blogoverflow.com/2014/08/23/binary-quadratic-forms-over-the-rational-integers-and-class-numbers-of-quadratic-%EF%AC%81elds/