On the First Cohomology of Local Units

The groups of units $U^i_L$ of a local field $L$ play an important role in algebraic number theory, especially in class field theoretic topics. Therefore, it is interesting to study these groups from a cohomological point of view. In this article, we study and compute the first cohomology of $U_L^1$, $U^2_L$ and $U^3_L$ under certain mild hypotheses, and discuss some results about general $U^i_L$'s.