Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Gauss in his book "Disquisitiones arithmeticae" considered only forms $ax^2+bxy+cy^2$ where $b$ is even, apparently because he had some notion of integral matrix in his mind even though he did not state it explicitly. This restriction has an advantage to present a form in Gauss's form with an integral matrix.

elements of $Sym^2\mathbb{Z}^2$ can be viewed naturally as forms $ax^2+bxy+cy^2$ where $a,b,c\in\mathbb{Z}$. Can we find such a representation like above for Gauss's form i.e some sort of symmetric power of a free module?

This question might be vague, but any interpretation might be useful for my purpose.

share|improve this question
-1: too vague. Are you asking for an explanation of the difference between symmetric bilinear forms and quadratic forms? –  S. Carnahan Nov 11 '10 at 13:59

1 Answer 1

Forms $ax^2+bxy+cy^2$ for which $a,b,c,\in \mathbb Z$ and $b$ is even correspond to the symmetric submodule of $(\mathbb Z^2)^{\otimes 2}$, i.e. the $S_2$ fixed elements of $(\mathbb Z x\oplus \mathbb Z y)^{\otimes 2}$.

Forms $ax^2+bxy+cy^2$ for which $a,b,c,\in \mathbb Z$ correspond to the symmetric quotient of $(\mathbb Z^2)^{\otimes 2}$, i.e. the quotient of $(\mathbb Z x\oplus \mathbb Z y)^{\otimes 2}$ by elements $a-\sigma(a)$ where $\sigma$ is the transposition of what is left and right of the tensor.

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.