On cohomology groups H¹ of G-modules of finite type over cyclic groups

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.