On cohomology groups $ H^{1} $ of $ G-$modules of finite type over cyclic groups

TL;DR

This paper investigates the first cohomology groups of finitely generated modules over cyclic groups, focusing on order 2 cases and applications to number theory.

Contribution

It provides explicit relations between cohomology group orders and invariants for cyclic groups of order 2, with applications to number fields and Pell equations.

Findings

Order of cohomology groups explicitly related to invariants for G of order 2

Applications to unit groups over quadratic extensions of number fields

Results used to study class numbers and Pell equations

Abstract

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.