Hilbert norm residue symbol
= Hilbert norm residue symbol
{c}
{title2=$(a,b)_{F,m}$}
For a <local field> $F$ containing $\mu_m$, the symbol is $\operatorname{rec}_F(b)(a^{1/m})/a^{1/m}\in\mu_m$, 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.