Kummer pairing
= Kummer pairing
{wiki=Kummer_theory}
When $\mu_n\subset K$, the Kummer pairing sends $\sigma\in\operatorname{Gal}(L/K)$ and $[x]\in K^*/(K^*)^n$ to $\sigma(\sqrt[n]{x})/\sqrt[n]{x}\in\mu_n$. Its elliptic analogue sends $\sigma$ and $[P]\in E(K)/nE(K)$ to $\sigma(Q)-Q\in E[n]$ for $nQ=P$.