Nonabelian norm-residue kernel theorem

For a finite Galois extension $L/K$ with group $G$, the kernel of the global reciprocity map $C_K\to G^{\mathrm{ab}}$ is $N_{L/K}{C}_L$. It follows that this norm group equals the norm group from the maximal abelian subextension $L^{[G,G]}/K$.