Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 125 / 4 / a / Solution

Let $G=\mathrm{Gal}(L/K)$. If $a\in K^{\times }$ becomes an $n$th power in $L$, choose $\alpha \in L$ with ${\alpha }^n=a$. Thenis a cocycle with values in the finite group ${\mu }_n(L)$. Changing $\alpha$ changes it by a coboundary, and its class is trivial exactly when $a$ was already an $n$th power in $K$. Thus the kernel injects into the finite group $H^1(G,{\mu }_n(L))$ and is finite. This is the multiplicative case of the finite-extension kernel of a Kummer map.