= Kummer pairing
Let $n$ be prime to $\operatorname{char}K$ and suppose $\mu_n\subset K$. For a finite <Galois extension> $L/K$, the multiplicative Kummer pairing has domain
$$
\operatorname{Gal}(L/K)\times V_L,\qquad V_L=(K^\times\cap L^{\times n})/(K^\times)^n,
$$
and sends $(\sigma,[a])$ to $\sigma(\alpha)/\alpha$, where $\alpha\in L$ and $\alpha^n=a$. The restriction to $V_L$ ensures that the chosen root actually lies in the domain of $\sigma$. Changing that root or the representative of $[a]$ leaves the value unchanged; multiplication in either argument multiplies the values. If $L$ is generated by these roots, this is the perfect duality of finite <Kummer theory>.<Fields and Galois Theory by James S. Milne>[Kummer theory, Theorem 5.30 and the preceding root-ratio homomorphism]{ref}
For an <elliptic curve>, choose $Q$ with $nQ=P$. The analogous expression is $\sigma Q-Q\in E[n]$. In general it is a <Galois cohomology> cocycle, satisfying $c(\sigma\tau)=c(\sigma)+\sigma c(\tau)$, rather than a <group homomorphism>. If all $n$-torsion is rational, the action on $E[n]$ is trivial and it becomes a homomorphism. This is the connecting construction in the <Kummer map of an elliptic curve>.
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