On generators of unramified Iwasawa module over an imaginary quadratic field

Let $p$ be an odd prime number and $k_p$ be the maximal pro-$p$ extension unramified outside $p$ of an imaginary quadratic field $k$. Let $\widetilde{k}$ be the maximal multiple $\mathbb{Z}_{p}$ extension over $k$ and $M_{\widetilde{k}}$ be the maximal abelian pro-$p$ extension unramified outside $p$ over $\widetilde{k}$. In this paper, we give an explicit description of a generating system of ${\rm{Gal}}(M_{\widetilde{k}}/\widetilde{k})$ and study its structure. As an application, we apply it to describe the unramified Iwasawa module $X_{k}$ for $k$.