On Certain Cohomological Invariants of Algebraic Number

We see the Poincare series from a cohomological point of view and apply the idea to a finite group G acting on any commutative ring R with unity. For a 1-cocycle c of G on the unit group R×, we define a |G|-torsion module Mc/Pc, which is independent of the choice of representatives of the cohomology class γ = [c]. We are mostly interested in determining Mc/Pc where G is the Galois group of a finite Galois extension K/k of algebraic number fields and R is the ring OK of integers in K. We determine Mc/Pc and the index iγ(K/k) = [Mc: Pc] in terms of the ramification index and the different DK/k. We will determine them explicitly for the case of quadratic, biquadratic, cyclotomic extensions, and the maximal real subfields of cyclotomic extensions over Q