Where the topology of Galois groups comes from?
问题内容
It is a well known fact that the Galois group $G$ of a Galois extension $K\subseteq L$ is a profinite group, as $G$ is equal to the inverse limit of the Galois groups of the finite subextensions of $K\subseteq L$. Therefore, $G$ gets a "natural" topology that turns it into a compact group.
Why is it the right topology to put on $G$? I mean, doesn't a Galois group admit other useful, or more natural topologies?
I read that Galois groups and fundamental groups are somewhat analogous objects (see a MO thread in here). However, fundamental groups arise at least at a first glance as objects without a clear topology... so what is different in the nature of Galois groups?
回答 (1)
I don't know how satisfying you'll find this, but the most obvious answer is that when $K\subseteq L$ is a finite Galois extension it does come with an obvious topology - the discrete topology.
Now suppose $K\subseteq L$ is an infinite Galois extension. The discrete topologies on the finite subextensions $K\subseteq L'\subseteq L$ "fit together" in the sense that the inverse system of groups is in fact an inverse system of topological groups. This is of course completely boring, but it suggests interpreting $\mathit{Gal}(L/K)$ as an inverse limit of topological groups, and now the discrete topologies on the "approximations" turn into an interesting (= profinite) topology on the result.
Also, re: topologizing the fundamental group, see e.g. Brazas' blog post on topologizing the fundamental group - note that one important point he makes is that when the underlying space is "nice" (e.g. a manifold) we can't hope to do any better than the discrete topology, which is in line with thinking of manifolds as finite Galois extensions per the idea above.