Definition (Quadratic form in an abelian group). Let $A$ be an abelian group. We say $q:A\to \mathbb{Z}$ is a quadratic form if

• (i) $q(nx)=n^{2}q(x)$ for all $n\in \mathbb{Z}$, $x\in A$.
• (ii) $(x,y)\mapsto q(x+y)-q(x)-q(y)$ is $\mathbb{Z}$-bilinear.