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I posted this question over on stack exchange and was told it would work better here. For interest I have been looking at links between class field theory and étale cohomology. Let $k$ be a global ...

Is it true that the group $$H^1(Gal(K^{ab}/K)/\mu_{\nu}(Gal(K_{\nu}^{ab}/K_{\nu})),E_{p^n})$$ is always p-divisible? Or are there any conditions which, when satisfied, guarantee its p-divisibility? ...