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 $k$.
Categories: math.NT