Let $K$ be a local field with $K^{ur}$ the maximal unramified extension and $K^{tr}$ the maximal tamely ramified extension. Assume that the characteristic of the residue field of $K$ is $q$.

If $p$ is a prime not equal to $q$, then we know that the cohomological dimension of $Gal (K^{ur}/K) = 1$ so $H^2 (Gal(K^{ur}/K), \mu_p) = 0$ as $Gal(K^{ur}/K) = \widehat{Z}$. Do we know if $H^2(Gal(K^{tr}/K), \mu_p) =0$ as well?

inj.$H^1(D/I,M) \rightarrow H^1(D,M))$. By Herbrand, $h^1(D/I,M) = h^0(D,M) = \#\mu_p(K)$. Since $H^1(D,M) = H^1(K,M)^P$ ($P=$ wild inertia) is $K^{\times}/(K^{\times})^p$ by Kummer, sizes differ. – BCnrd Sep 14 '10 at 15:27