Quadratic Hilbert symbol
= Quadratic Hilbert symbol
{title2=$(a,b)_v\in\{1,-1\}$}
The symbol is one exactly when $b$ is a norm from $K_v(\sqrt a)$, and is always one if $a$ is square. Equivalently the conic $z^2=ax^2+by^2$ is isotropic. This proves symmetry. The <cyclic local norm index> makes the norm subgroup have index two for nonsquare $a$, proving bilinearity and nondegeneracy on local square classes. Also $(a,-a)_v=1$ and $(a,a)_v=(a,-1)_v$.