On the Finiteness and Structure of Galois Groups of Tamely ramified pro-p Extensions of Imaginary Quadratic Fields

TL;DR

This paper investigates the structure and finiteness of Galois groups of maximal pro-p extensions of imaginary quadratic fields, providing explicit presentations and analyzing cases for p odd, especially for p=3.

Contribution

It offers explicit presentations of Galois groups for certain imaginary quadratic fields and extends understanding of their structure in the context of tamely ramified pro-p extensions.

Findings

Explicit presentations of Galois groups for odd p under certain conditions

Determination of the structure of the maximal pro-3 extension of Q(i)

Insights into the finiteness properties of these Galois groups

Abstract

For a prime p, we study the Galois groups of maximal pro-$p$ extensions of imaginary quadratic fields unramified outside a finite set $S$, where $S$ consists of one or two finite places not lying above $p$. When $p$ is odd, we give explicit presentations of these Galois groups under certain conditions. As an application, we determine the structure of the maximal pro-$3$ extension of $\mathbb{Q}(i)$ unramified outside two specific finite places.