I’m studying Iwasawa theory and I meet some questions Thanks a lot for your help.

Q_1: About terminology $p$-extension. I find many reference use maximal $p$-extension or maximal abelian p-extension of a cyclotomic $\mathbb{Z}_p$ extension $F_\infty/F$. I know actually it is the inverse limit of class group, so it is a pro-$p$ group, isn’t it? Does it means in many cases our maximal $p$-extension, actually which may be not an exactly $p$-extension, is a pro-$p$ extension? And also, is it very common in many reference pro-p extension and $p$-extension are used chaos?

By the way, Let cyclotomic tower $F_n$ of $F$, $L_n$ $p$-Hilbert field of $F_n$, is the union of $L_n$=$L_\infty$ maximal abelian unramified pro-$p$ extension of $F_\infty$？

Q_2: I also notice maximal pro-$p$ extension occurring in many non-commutative Iwasawa theory cases. If we fix a number field $F$, I think maximal pro-$p$ and maximal $p$-extension are different? Also, how could we know the existence of the maximal pro-$p$ extension?

Another question: On Iwasawa algebra of $\mathbb{Z}_p$, here it has coefficients in $\mathbb{Z}_p$, we know there could be three topology, $p,T,m=（p,T)$-adic resp. It seems the three are the same in $\Lambda(\mathbb{Z}_p)$? When we consider the $\Lambda(\mathbb{Z}_p)$-action on $X=\mathrm{Gal}(L_\infty/F_\infty)$, it is said $X$ is compact module. Is the topology on $X$ inverse topology of profinite group? How could we see the multiplication of $\Lambda(\mathbb{Z}_p)$ on $X$ is continuous?

Which cases the three are not the same? Like if we permit the coefficients be in general integer ring of local field? Or the group $G$ in $\Lambda(G)$ is non-commutative?

Thanks again for your help!

I’m studying Iwasawa theory and I meet some questions Thanks a lot for your help.

Q_1: About terminology $p$-extension. I find many reference use maximal $p$-extension or maximal abelian p-extension of a cyclotomic $\mathbb{Z}_p$ extension $F_\infty/F$. I know actually it is the inverse limit of p-part of class group, so it is a pro-$p$ group, isn’t it? Does it means in many cases our maximal $p$-extension, actually which may be not an exactly $p$-extension, is a pro-$p$ extension? And also, is it very common in many reference pro-p extension and $p$-extension are used chaos?

By the way, Let cyclotomic tower $F_n$ of $F$, $L_n$ $p$-Hilbert field of $F_n$, is the union of $L_n$=$L_\infty$ maximal abelian unramified pro-$p$ extension of $F_\infty$？

Q_2: I also notice maximal pro-$p$ extension occurring in many non-commutative Iwasawa theory cases. If we fix a number field $F$, I think maximal pro-$p$ and maximal $p$-extension are different? Also, how could we know the existence of the maximal pro-$p$ extension?

Another question: On Iwasawa algebra of $\mathbb{Z}_p$, here it has coefficients in $\mathbb{Z}_p$, we know there could be three topology, $p,T,m=（p,T)$-adic resp. It seems the three are the same in $\Lambda(\mathbb{Z}_p)$? When we consider the $\Lambda(\mathbb{Z}_p)$-action on $X=\mathrm{Gal}(L_\infty/F_\infty)$, it is said $X$ is compact module. Is the topology on $X$ inverse topology of profinite group? How could we see the multiplication of $\Lambda(\mathbb{Z}_p)$ on $X$ is continuous?

Which cases the three are not the same? Like if we permit the coefficients be in general integer ring of local field? Or the group $G$ in $\Lambda(G)$ is non-commutative?

Thanks again for your help!

I’m studying Iawasawa theory and I meet some questions Thanks a lot for your help.

Q_1: About terminology p-extension. I find many reference use maximal p-extension or maximal abelian p-extension of a cyclotomic Z_p extension $F_\infty/F$. I know actually it is the inverse limit of class group, so it is a pro-p group, isn’t it? Does it means in many cases our maximal p-extension, actually which may be not an exactly p-extension, is a pro-p extension? And also, is it very common in many reference pro-p extension and p-extension are used chaos?

By the way, Let cyclotomic tower $F_n$ of $F$, $L_n$ p-Hilbert field of $F_n$, is the union of $L_n$=$L_\infty$ maximal abelian unramified pro-p extension of $F_\infty$？

Q_2: I also notice maximal pro-p extension occurring in many non-commutative Iwasawa theory cases. If we fix a number field F, I think maximal pro-p and maximal p-extension are different? Also, how could we know the existence of the maximal pro-p extension?

Another question: On Iwasawa algebra of Z_p, here it has coefficients in $Z_p$, we know there could be three topology, p,T,m=（p,T)-adic resp. It seems the three are the same in $\Lambda(Z_p)$? When we consider the $\Lambda(Z_p)$ action on X=Gal($L_\infty/F_\infty$), it is said X is compact module. Is the topology on X inverse topology of profinite group? How could we see the multiplication of $\Lambda(Z_p)$ on X is continuous?

Which cases the three are not the same? Like if we permit the coefficients be in general integer ring of local field? Or the group G in $\Lambda(G)$ is non-commutative?

Thanks again for your help!

I’m studying Iwasawa theory and I meet some questions Thanks a lot for your help.

Q_1: About terminology $p$-extension. I find many reference use maximal $p$-extension or maximal abelian p-extension of a cyclotomic $\mathbb{Z}_p$ extension $F_\infty/F$. I know actually it is the inverse limit of class group, so it is a pro-$p$ group, isn’t it? Does it means in many cases our maximal $p$-extension, actually which may be not an exactly $p$-extension, is a pro-$p$ extension? And also, is it very common in many reference pro-p extension and $p$-extension are used chaos?

By the way, Let cyclotomic tower $F_n$ of $F$, $L_n$ $p$-Hilbert field of $F_n$, is the union of $L_n$=$L_\infty$ maximal abelian unramified pro-$p$ extension of $F_\infty$？

Q_2: I also notice maximal pro-$p$ extension occurring in many non-commutative Iwasawa theory cases. If we fix a number field $F$, I think maximal pro-$p$ and maximal $p$-extension are different? Also, how could we know the existence of the maximal pro-$p$ extension?

Another question: On Iwasawa algebra of $\mathbb{Z}_p$, here it has coefficients in $\mathbb{Z}_p$, we know there could be three topology, $p,T,m=（p,T)$-adic resp. It seems the three are the same in $\Lambda(\mathbb{Z}_p)$? When we consider the $\Lambda(\mathbb{Z}_p)$-action on $X=\mathrm{Gal}(L_\infty/F_\infty)$, it is said $X$ is compact module. Is the topology on $X$ inverse topology of profinite group? How could we see the multiplication of $\Lambda(\mathbb{Z}_p)$ on $X$ is continuous?

Which cases the three are not the same? Like if we permit the coefficients be in general integer ring of local field? Or the group $G$ in $\Lambda(G)$ is non-commutative?

Thanks again for your help!