The Kummerian Property and Maximal Pro-$p$ Galois Groups

TL;DR

This paper introduces a new restriction on pro-$p$ groups that can be realized as maximal pro-$p$ Galois groups of fields containing a $p$th root of unity, based on Kummer Theory and cohomological methods.

Contribution

It provides a novel restriction criterion for pro-$p$ Galois groups using Kummer Theory and characterizes it cohomologically, leading to new examples of non-realizable groups.

Findings

Identifies a new restriction on pro-$p$ Galois groups

Characterizes the restriction cohomologically

Constructs examples of groups not realizable as Galois groups

Abstract

For a prime number $p$, we give a new restriction on pro-$p$ groups $G$ which are realizable as the maximal pro-$p$ Galois group $G_F(p)$ for a field $F$ containing a root of unity of order $p$. This restriction arises from Kummer Theory and the structure of the maximal $p$-radical extension of $F$. We study it in the abstract context of pro-$p$ groups $G$ with a continuous homomorphism $\theta\colon G\to1+p\mathbb{Z}_p$, and characterize it cohomologically, and in terms of 1-cocycles on $G$. This is used to produce new examples of pro-$p$ groups which do not occur as maximal pro-$p$ Galois groups of fields as above.