November 1st
Today I learned the definition of the Krull topology. Basically, the idea is that we want to have some nicer control over infinite Galois extensions, and there's a notion of "closeness'' provided between automorphisms, so this will induce a topology. More formally, fix a probably infinite extension $L/K.$ Then for an intermediate field $K'$ where $K'/K$ is finite, we define the neighborhood\[U_{\sigma,K'}=\{\tau\in\op{Gal}(L/K):\tau=\sigma\text{ on }K'\}.\]So two automorphisms are "close'' if they agree on a "small'' subfield $K'/K.$ We can also write\[U_{\sigma,K'}=\sigma\left\{\sigma^{-1}\tau\in\op{Gal}(L/K):\sigma^{-1}\tau\text{ fixes }K'/K\right\},\]so our open sets are actually cosets of $\op{Aut}(K'/K).$ Note $K'/K$ is not required to be Galois, so this might not even be a coset of a subgroup.
I guess most pedestrian examples aren't super interesting. For example, any finite extension $L/K$ will just get the discrete topology. Indeed, for some $\sigma\in\op{Gal}(L/K),$ we get that\[U_{\sigma,L}=\{\tau:\tau=\sigma\text{ on }L\}=\{\sigma\},\]which is legal because $L/K$ is a finite extension. It follows that every singleton is open, giving us the discrete topology.
With $\op{Gal}(\QQ(\zeta_{p^\infty})/\QQ),$ we see that our open sets are really\[U_{\sigma,\QQ(\zeta_{p^\bullet})}=\sigma\cdot\op{Gal}(\QQ(\zeta_{p^\bullet})/\QQ)=a_\sigma\ZZ_p^\times,\]where $a_\sigma$ is our element of $\ZZ_p^\times$ corresponding to the automorphism $\sigma.$ It follows that the topology over $\ZZ_p$ more or less caries over with the obvious caveat that we're living in $\ZZ_p^\times,$ provided that I didn't make a mistake somewhere. Basically, we're finding that $\ZZ_p^\times$ and $\op{Gal}(\QQ(\zeta_{p^\infty})/\QQ)$ are not just the same as groups, but it's nice to note that the natural isomorphism can also preserve the topology of $\ZZ_p.$