Let $G=\operatorname{Gal}(L/K)$. If $a\in K^\times$ becomes an $n$th power in $L$, choose $\alpha\in L$ with $\alpha^n=a$. Then
$$
\sigma\longmapsto\frac{\sigma(\alpha)}{\alpha}
$$
is 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>.

Solution

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

For a finite Galois extension $L/K$, the kernel of
$$
K^*/(K^*)^n\longrightarrow L^*/(L^*)^n
$$
injects into the finite group $H^1(\operatorname{Gal}(L/K),\mu_n(L))$. Likewise, the kernel of $E(K)/nE(K)\to E(L)/nE(L)$ injects into $H^1(\operatorname{Gal}(L/K),E[n](L))$ and is finite.


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