On the First Cohomology of Local Units

TL;DR

This paper investigates the first cohomology groups of local units in local fields, providing explicit computations for certain cases and discussing broader implications in algebraic number theory.

Contribution

It offers new cohomological calculations for specific unit groups in local fields and explores general properties of these groups.

Findings

Computed first cohomology of $U_L^1$, $U_L^2$, and $U_L^3$

Discussed properties of general $U_L^i$ groups

Enhanced understanding of local units in cohomological context

Abstract

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.