R\'edei symbols and arithmetical mild pro-2-groups

TL;DR

This paper describes the triple Massey product for certain Galois groups using Rédéi symbols, showing the existence of specific mild pro-2-groups with particular properties.

Contribution

It provides an explicit description of the triple Massey product for Galois groups and constructs examples of mild pro-2-groups with Zassenhaus invariant 3.

Findings

Explicit description of triple Massey product via Rédéi symbols

Existence of mild pro-2-groups with Zassenhaus invariant 3 as Galois groups

Construction of a non-analytic mild fab pro-2-group with 3 generators

Abstract

Generalizing results of Morishita and Vogel, an explicit description of the triple Massey product for the Galois group $G_S(2)$ of the maximal 2-extension of $\mathbb{Q}$ unramified outside a finite set of prime numbers $S$ containing 2 is given in terms of R\'edei symbols. We show that mild pro-2-groups with Zassenhaus invariant 3 occur as Galois groups of the form $G_S(2)$. Furthermore, a non-analytic mild fab pro-2-group having only 3 generators is constructed .