Hilbert norm residue symbol ($(a,b{)}_{F,m}$)

For a local field $F$ containing ${\mu }_m$, the symbol is ${\mathrm{rec}}_F(b)(a^{1/m})/a^{1/m}\in {\mu }_m$, after fixing a convention for Local Artin reciprocity. It is independent of the chosen root and bilinear. Its kernel in the second variable is the norm subgroup of $F(a^{1/m})/F$. The quadratic Hilbert symbol is the sign-valued case relevant to quadratic forms.

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 $z^2=a{x}^2+b{y}^2$ 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 $(a,a{)}_v=(a,-1{)}_v$.

For nonzero $a,b$ in a number field, the product of their quadratic Hilbert symbols over all places is one. All but finitely many symbols are one. Over $\mathbb{Q}$, this is equivalent to quadratic reciprocity with its supplementary laws. It supplies the compatibility relation between the local invariants of a global quadratic form.