Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 123 / 4 / 2 / d / Solution

Let $G=\mathrm{Gal}(L/K)$ and letbe the fixed field of the commutator subgroup. Then $M/K$ is the maximal abelian subextension of $L/K$, withThe nonabelian norm-residue kernel theorem identifies the kernel of the global reciprocity mapwith $N_{L/K}{C}_L$. Applied to the abelian extension $M/K$, the Artin reciprocity law identifies the kernel of the same map with $N_{M/K}{C}_M$. Thereforeso the norm group of the Galois extension $L/K$ is the norm group of the abelian extension $M/K$.