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

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.