Demushkin groups of uncountable rank

TL;DR

This paper generalizes the theory of Demushkin groups from countable to uncountable rank, exploring their algebraic properties, classification, and connections to Galois groups, including their profinite completions.

Contribution

It extends the theory of Demushkin groups to uncountable rank, analyzing their invariants, classification, and realization as Galois groups, and computes their profinite completions.

Findings

Classification of uncountable rank Demushkin groups

Calculation of their profinite completions

Results on the profinite completion of absolute Galois groups

Abstract

We extend the theory of countably generated Demushkin groups to Demushkin groups of arbitrary rank. We investigate their algebraic properties and invariants, count their isomorphism classes and study their realization as absolute Galois group. At the end, we compute their profinite completion and conclude with some results on profinite completion of absolute Galois groups.