Let $k$ be a field that is finitely-generated over its prime field and $\bar k$ a separable closure. I would like to compute
$$ H^1_{cts}(\mathrm{Gal}_k, \hat{\mathbb{Z}}(1)) $$
Since $\mathbb{Z}(1) = \varprojlim \mu_{n, \bar k}$, we have
$$ H^1_{cts}(\mathrm{Gal}_k, \hat{\mathbb{Z}}(1)) = \varprojlim_n H^1(\mathrm{Gal}_k, \mu_{n, \bar k}) = \varprojlim_n k^{\times} / (k^{\times})^n $$ Is there anything I can say about this inverse limit?