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For $[x]\in \Delta$, choose ${\alpha }_x$ with ${\alpha }_x^n=x$ and define the Kummer pairingChanging $x$ by an $n$th power or changing its root leaves the quotient unchanged because ${\mu }_n\subset K$. Multiplication of radicals proves bilinearity. If an automorphism pairs trivially with every class, it fixes all generating radicals and hence all of $L$. If a class pairs trivially with every automorphism, its radical lies in $K$, so the class is trivial. Thus the pairing is well-defined, bilinear, and nondegenerate on both sides.

Solved by gpt-5.6-sol high.

For $[P]\in \Gamma$, choose $Q$ with $nQ=P$ and define the elliptic Kummer pairingReplacing $Q$ by $Q+T$ with $T\in E[n]$ changes nothing because all $n$-torsion is $K$-rational; replacing $P$ by $P+nR$ permits replacing $Q$ by $Q+R$, again without changing the value. The group laws prove bilinearity. An automorphism pairing trivially with every class fixes every division point and hence $L$. Conversely, if $[P]$ pairs trivially with every automorphism, a chosen $Q$ is Galois fixed, so $Q\in E(K)$ and $P=nQ$. The pairing is therefore nondegenerate.

Solved by gpt-5.6-sol high.