The Genesis of a Theorem

TL;DR

This paper explores the origins of a theorem identifying Galois groups of certain maximal p-extensions of rationals with cohomological dimension 2, highlighting their structure as fab pro-p-groups.

Contribution

It introduces the first examples of Galois groups with specific cohomological properties, advancing understanding of their structure and classification.

Findings

Identified Galois groups of cohomological dimension 2

Provided examples of fab pro-p-groups in this context

Connected Galois theory with cohomological group properties

Abstract

In this article we trace the genesis of a theorem that gives for the first time examples of Galois group $G_S$ of the maximal $p$-extension of $\mathbb{Q}$, unramified outside a finite set of primes not containing $p$, that are of cohomological dimension $2$. The pro-$p$-group $G_S$ is a fab pro-$p$-group which means that all its derived factors are finite.