= Anisotropic quadratic form
{title2=$q(v)=0\Longrightarrow v=0$}
A <quadratic form> $q$ over a <field> $k$ is an <anisotropic quadratic form> if $q(v)=0$ forces $v=0$. An <isotropic quadratic form> instead has a nonzero zero vector. This is a property of the form together with its coefficient field, rather than just the displayed polynomial.<Introduction to quadratic forms by Bill Casselman>[Section 2, isotropic and anisotropic vectors, printed page 5]{ref}
For example, $q(x,y)=x^2-2y^2$ is anisotropic over $\mathbb Q$. If a nonzero rational solution had $y=0$, then $x=0$ too; otherwise $x/y$ would be a rational square root of two. Over $\mathbb R$, however, $(\sqrt2,1)$ is a nonzero zero vector. The coordinate change $u=x+\sqrt2y$, $v=x-\sqrt2y$ gives $q=uv$, a <hyperbolic plane of a quadratic form>.
This example also separates anisotropy from <positive definiteness>: the rational form takes the positive value $1$ and the negative value $-2$, despite having no nonzero rational zero. In a <Witt decomposition>, an anisotropic remainder is what remains after all hyperbolic planes have been split off. Its <Witt index> is zero. In characteristic two, one must test vanishing of the quadratic form itself; properties of the associated polar <bilinear form> alone do not supply the same test.
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