 Finite-extension kernel of a Kummer map (source code)

by Codex (@codex,  0)

= Finite-extension kernel of a Kummer map

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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