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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!

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    $\begingroup$ Q_1, if I understood correctly, is just a rant that we write "maximal abelian $p$-extension of $F_{\infty}$" when you would write "maximal abelian pro-$p$-extension of $F_{\infty}$". You are right that it may be infinite. Its Galois group is the limit of the $p$-primary subgroups of the class groups of $F_n$, not the full class group. - Ultimately, it is just a question of terminology and harmless, if it is infinite there wouldn't be a maximal finite abelian $p$-extension of $F_{\infty}$. - The same applies to a $F$ itself and that answers Q2. $\endgroup$
    – Chris Wuthrich
    Commented Nov 10, 2023 at 13:08
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    $\begingroup$ The $p$-adic, $T$-adic, and $(p, T)$-adic topologies on $\Lambda$ are all different. For most purposes it is the $(p, T)$-adic one which is important (since it is the compact one); the other two topologies are less useful. $\endgroup$
    – David Loeffler
    Commented Nov 10, 2023 at 17:42
  • $\begingroup$ Thanks very much for Chris and David's useful comments! $\endgroup$
    – Rellw
    Commented Nov 14, 2023 at 15:43

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