 Finite-extension kernel of a Kummer map

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