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In Tate's local duality theorem we find the isomorphism $H^2(K, Q_p(1)) \cong Q_p$ where $K$ is a finite extension of $Q_p$. I haven't found any reference where this isomorphism is given explicitly (Only Kato's article in "Grothendieck Festchrift). Can anyone give me some reference or explicit this isomorphism please?

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Provided you know what whatever definition you use agrees with inverting $p$ upon the inverse-limit definition applied to a Galois lattice, you want to identify ${\rm{H}}^2(K,\mathbf{Z}_p(1))= \invlim {\rm{H}}^2(K,\mu_{p^n})$ with $\mathbf{Z}_p$, or more specifically to show ${\rm{H}}^2(K,\mu_{p^n})\simeq \mathbf{Z}/(p^n)$ compatibly with change in $n$ ("$p$-mult." on left, reduction on right). Since ${\rm{Br}}(K)=\mathbf{Q}/\mathbf{Z}$, pass to $p$-power torsion and look at $p$-multiplication from $p^{n+1}$-torsion into $p^n$-torsion, and relate Kummer sequences for $p^n$ and $p^{n+1}$, etc. – user29720 Jun 14 '13 at 13:14

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