Let $p$ be a prime number and $H$ be number field. Denote by $\tilde{H}$ the composite of all $\mathbb{Z}_p$ extensions of $H$, so $Gal(\tilde{H}/H)=\mathbb{Z}_p^{d}$. Write $\varLambda$ for $\mathbb{Z}_p [[Gal(\tilde{H}/H)]]\simeq\mathbb{Z}_p[[t_1, ... t_d]]$. Let $M$ be the maximal abelian unramified $p$-extension of $\tilde{H}$, as well known $X=Gal(M/\tilde{H})$ is noetherian torsion module over $\varLambda$.

1)Is $\frac{X}{(t_1,t_2,...,t_n)X}$ of finite cardinal or torsion group?

Consider $L$ the fixed field by $(t_1,t_2,...,t_n)X$, so $Gal(L/\tilde{H})=\frac{X}{(t_1,t_2,...,t_n)X}$.

2)Is $L$ an abelian extension of $H$.

Let $p$ be a prime number and $H$ be number field. Denote by $\tilde{H}$ the composite of all $\mathbb{Z}_p$ extensions of $H$, so $\operatorname{Gal}(\tilde{H}/H)=\mathbb{Z}_p^{d}$. Write $\varLambda$ for $\mathbb{Z}_p [[\operatorname{Gal}(\tilde{H}/H)]]\simeq\mathbb{Z}_p[[t_1, ... t_d]]$. Let $M$ be the maximal abelian unramified $p$-extension of $\tilde{H}$, as well known $X=\operatorname{Gal}(M/\tilde{H})$ is noetherian torsion module over $\varLambda$.

1. Is $\frac{X}{(t_1,t_2,...,t_n)X}$ of finite cardinal or torsion group?

Consider $L$ the fixed field by $(t_1,t_2,...,t_n)X$, so $\operatorname{Gal}(L/\tilde{H})=\frac{X}{(t_1,t_2,...,t_n)X}$.

2. Is $L$ an abelian extension of $H$.

Let $p$ be a prime number and $H$ be number field. Denote by $\tilde{H}$ the composite of all $\mathbb{Z}_p$ extensions of $H$, so $Gal(\tilde{H}/H)=\mathbb{Z}_p^{d}$. Write $\varLambda$ for $\mathbb{Z}_p [[Gal(\tilde{H}/H)]]\simeq\mathbb{Z}_p[[t_1, ... t_d]]$. Let $M$ be the maximal abelian unramified $p$-extension of $\tilde{H}$, as well known $X=Gal(M/\tilde{H})$ is noetherian torsion module over $\varLambda$.

1)Is $\frac{X}{(t_1,t_2,...,t_n)X}$ of finite cardinal?

Consider $L$ the fixed field by $(t_1,t_2,...,t_n)X$, so $Gal(L/\tilde{H})=\frac{X}{(t_1,t_2,...,t_n)X}$.

2)Is $L$ an abelian extension of $H$.

Let $p$ be a prime number and $H$ be number field. Denote by $\tilde{H}$ the composite of all $\mathbb{Z}_p$ extensions of $H$, so $Gal(\tilde{H}/H)=\mathbb{Z}_p^{d}$. Write $\varLambda$ for $\mathbb{Z}_p [[Gal(\tilde{H}/H)]]\simeq\mathbb{Z}_p[[t_1, ... t_d]]$. Let $M$ be the maximal abelian unramified $p$-extension of $\tilde{H}$, as well known $X=Gal(M/\tilde{H})$ is noetherian torsion module over $\varLambda$.

1)Is $\frac{X}{(t_1,t_2,...,t_n)X}$ of finite cardinal or torsion group?

Consider $L$ the fixed field by $(t_1,t_2,...,t_n)X$, so $Gal(L/\tilde{H})=\frac{X}{(t_1,t_2,...,t_n)X}$.

2)Is $L$ an abelian extension of $H$.