On the abelian groups which occur as Galois cohomology groups of global unit groups

TL;DR

This paper characterizes which abelian groups appear as Galois cohomology groups of global unit groups in unramified G-extensions of number fields, focusing on finite p-groups for specific cohomological degrees.

Contribution

It determines the sets of possible Galois cohomology groups for degrees 0, 1, 2, and 4 when G is a finite p-group, providing new classifications.

Findings

Identifies all isomorphism classes of Galois cohomology groups for specified degrees.

Provides explicit descriptions for p-group cases.

Advances understanding of the structure of global units in Galois extensions.

Abstract

For any finite group G and integer i, let $\mathcal{H}^i(G)$ be the set of all the isomorphism classes of the Galois cohomology groups $\hat{H}^i(K/k,E_K)$, where K/k runs over all the unramified G-extension of number fields and E_K denotes the global unit group of K. We will determine $\mathcal{H}^i(G)$ for i=0,1,2, and 4 in the case where G is a finite p-group.