Show that for any local field K, there is a finite Galois extension L, such that $H^{i}_{T}(Gal(L/K),O_L^*)$ does not vanish for all i, here $O_L$ is ring of algebraic integers of L.
I only know that for unramified extension L/K, such cohomology always vanish. I do not know how to construct one with converse property.