  Let $ G $ be a cyclic group, in this paper, we study the Herbrand quotient
and $ 1-$th cohomology group on finitely generated $ G-$modules in some cases.
When $ G $ is of order $ 2, $ the order of the cohomology group is explicitly
related to some invariants, and this relation is used to study unit groups over
quadratic extensions of number fields. We also give some applications on Pell
equations and class number of number fields.
