Quotients of absolute Galois groups which determine the entire Galois cohomology

TL;DR

This paper demonstrates that the Galois cohomology ring of a field with certain roots of unity is fully determined by a specific quotient of its absolute Galois group, and vice versa, leading to new insights into which pro-p groups can be Galois groups.

Contribution

It establishes a precise relationship between a quotient of the Galois group and the Galois cohomology ring, providing new examples of pro-p groups not realizable as Galois groups.

Findings

Galois cohomology ring determined by a quotient of the Galois group

The quotient G_F^{[3]} is uniquely determined by the lower cohomology

New examples of pro-p groups not arising as Galois groups

Abstract

For prime power $q=p^d$ and a field $F$ containing a root of unity of order $q$ we show that the Galois cohomology ring $H^*(G_F,\dbZ/q)$ is determined by a quotient $G_F^{[3]}$ of the absolute Galois group $G_F$ related to its descending $q$-central sequence. Conversely, we show that $G_F^{[3]}$ is determined by the lower cohomology of $G_F$. This is used to give new examples of pro-$p$ groups which do not occur as absolute Galois groups of fields.