# quadratic forms

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• May 3rd 2010, 10:52 AM
EinStone
quadratic forms
Consider a quadratic form $f(x,y) = ax^2 + bxy + cy^2$ with $a,b,c \in \mathbb{Z}$ and discriminant $D = b^2 - 4ac$

We know that f factors as a product of linear forms $(ux+vy)(u'x+v'y)$ with rational coefficients if and only if D is a square.
Under which conditions for $a,b,c$ are $u,v,u',v' \in \mathbb{Z}$?
• May 3rd 2010, 05:14 PM
Bruno J.
You might want to use Gauss's Lemma.
• May 4th 2010, 12:48 PM
EinStone
I don't get Gauss lemma and I don't see how to use it here.