Mild pro-$p$-groups and $p$-extensions of imaginary quadratic fields with non-trivial $p$-class group

TL;DR

This paper investigates conditions under which the Galois group of certain maximal pro-$p$-extensions of imaginary quadratic fields with specific class group properties is mild, implying it has cohomological dimension two.

Contribution

It provides sufficient conditions for the Galois group to be mild in the context of imaginary quadratic fields with a p-class group of rank one.

Findings

Galois group G_S can be mild under certain conditions

Mild Galois groups have cohomological dimension 2

Results apply to fields with specific class group structures

Abstract

Let $k$ be an imaginary quadratic field and $p$ an odd prime number such that the $p$-rank of the class group of $k$ is one. Let $S$ be a finite set of places of $k$ distinct from $p$-adic places. We give sufficient conditions for the Galois group $G_S$, of the maximal pro-$p$-extension of $k$ which is unramified outside $S$, to be \textit{mild}, hence of cohomological dimension $2$.