= Solution
The translated <cyclotomic polynomial>
$$
\Phi_{p^n}(1+T)=\frac{(1+T)^{p^n}-1}{(1+T)^{p^{n-1}}-1}
$$
is Eisenstein at $p$. Hence $\zeta_{p^n}-1$ generates a totally ramified extension of degree
$$
\varphi(p^n)=p^{n-1}(p-1).
$$
Every automorphism sends $\zeta_{p^n}$ to $\zeta_{p^n}^a$ for a unique $a\in(\mathbb Z/p^n\mathbb Z)^\times$. This gives an injection
$$
\psi_n:\operatorname{Gal}(K_n/\mathbb Q_p)\longrightarrow(\mathbb Z/p^n\mathbb Z)^\times,
$$
and equality of orders makes it an isomorphism.
Normalize the <Local Artin map> so that $p$ maps to arithmetic Frobenius on the maximal unramified extension. For $u\in\mathbb Z_p^\times$, define its action on $K_n$ by
$$
\operatorname{Art}_{\mathbb Q_p}(u)(\zeta_{p^n})=\zeta_{p^n}^{u^{-1}\bmod p^n},
$$
where using $u$ rather than $u^{-1}$ gives the opposite Frobenius convention. These compatible maps, together with the image of $p$, define reciprocity on $p^{\mathbb Z}\times\mathbb Z_p^\times=\mathbb Q_p^\times$ and hence on every finite abelian extension by the <Local Kronecker-Weber theorem>.
Solved by gpt-5.6-sol high.
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