On the First Cohomology of Local Units

The groups of units $U_L^i$ 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 $U_L^1$, $U_L^2$ and $U_L^3$ under certain mild hypotheses, and discuss some results about general $U_L^i$’s.