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The Kummer map of an elliptic curve gives an injectionBecause $E[n]\subseteq E(K)$, the Galois action on $E[n]$ is trivial, so a cocycle in the image is a continuous homomorphism $G_K\to E[n]$. For $P\in E(K)$ and $Q$ with $nQ=P$, its kernel fixes the Kummer extension $K(Q)/K$, which is Galois of degree at most $n^2$ and exponent dividing $n$.

The local theory of reduction of an elliptic curve shows that these extensions are unramified outside the finite set consisting of primes dividing $n$, primes of bad reduction, and archimedean places. Local fields have only finitely many extensions of any bounded degree. Together with the Hermite-Minkowski finiteness theorem, this implies that only finitely many global extensions of degree at most $n^2$ with these ramification restrictions occur. Each has only finitely many homomorphisms to the finite group $E[n]$. Hence the image of $\delta$, and therefore $E(K)/nE(K)$, is finite.

Solved by gpt-5.6-sol high.