Symmetric Power and Gauss quadratic form

Question

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.

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.