In infinite Galois theory,main theorem failed and we get a "Krull topology" to mend the main theorem, we even generalized that to make definition for profinite group.

In local class field theory, the multiplicative group modulo the norm group is isomorphic to the Galois group both algebraically and topologically. So is the topology we just defined on infinite Galois group is "forced to define" because of that isomorphism? If you teach such a course in Galois theory, how would you convince your students that this topology comes out in "natural"?

There is another way topology arise: by study prime $p$, we deduce the $p$-adic topology, we complete the space we get local field and also a topological field. Can we realize that all the topologies that arise in number theory are just a mimic of that?

Since I am not a math major, I get into math since my teacher teach me quadratic reciprocity when I was a high schooler, I led all the way myself to class field theory and get lost when I encounter the "modern language" such as Galois cohomology and schemes. I saw a remark that in order to gain arithmetic information we need a ground base: the direct product of all $\mathbb{Q}_p$ ($p$-adic number field), but since it is not locally compact, we can not apply analysis on it, so we make restricted product and get idèles and adèles.

I know little about analysis and I get puzzled. Hecke proved quadratic reciprocity for number field using Fourier analysis and Tate has a famous "Thesis", both of which confused me.

Can someone explain that to me, what led these guys to do that,and what were they doing?

Finally I summarize my odd words:

Why we introduce topology to various group such as a Galois group, and how do we realize that topology?

Why, in doing arithmetic, do we need a locally compact group or field to apply "analysis"? what is "apply analysis" all about? (If it is not that easy to explain for a outsider, give me a link to some elementary resource as well; thanks!)