= Solution
For $[x]\in\Delta$, choose $\alpha_x$ with $\alpha_x^n=x$ and define the <Kummer pairing>
$$
\langle\sigma,[x]\rangle=\frac{\sigma(\alpha_x)}{\alpha_x}\in\mu_n.
$$
Changing $x$ 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.
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