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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:

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

  2. 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!)

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It seems like you answered your own first question -- the topology on profinite Galois groups certainly isn't forced by class field theory, but it's forced if you want Galois theory to work right. Specifically, the fundamental of Galois theory (the bijection between subextensions and subgroups) fails for infinite Galois groups considered as discrete groups, but is fixed by topologizing the subgroups via the Krull topology. For your other question, if you search this site, you'll see a nice answer to a question on Tate's thesis by Matt E which is well worth reading. –  Cam McLeman Jan 31 '12 at 5:12
    
did you mean we define closed sets just the topology that make Galois main theorem work?Like in algebraic geometry,we study Zar-topology just defined to link with polynomial functions,so that topology is the same thing with polynomials and their defining domain,just a way of speaking mimic general topology such as in a manifold? –  Yoshinobu Osawa Feb 1 '12 at 4:25

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I think a basic observation is that one can obtain a topology on a ring $R$ simply from the data of an ideal $I \subset R$; this is the $I$-adic topology, where a neighborhood basis at the identity consists of the powers of the ideals $I^i$. The nice thing about doing so is that, in essence, solving a polynomial equation mod a high power of $I$ essentially amounts to finding an "approximate solution" in $R$ itself: the notion of "approximate" solution is made valid by the existence of a topology.

In the case where $I $ is the ideal $p \mathbb{Z}$ inside the ring $\mathbb{Z}$, then the point of this discussion is that one has translated (purely arithmetic) questions of the form "When does there exist an integer $x$ such that $P(x) \equiv 0 \mod p^N$?" into more analysis-y questions of the form "When does there exist an integer $x$ such that $P(x) $ is close to zero?" So far, we have just moved words around without content, but we can make a big step forward by noting that analysis actually works pretty well only when one has a completeness property (like the completeness property of the real numbers). Without the completeness property of the real numbers, most of ordinary mathematical analysis would fail: try doing integration over the rationals, for instance.

Anyway, one does not usually have completeness of the topology induced on the ring $R$ given by an ideal $I$, but there is a mathematical procedure that enables you to replace $R$ by something where the completeness process is true: this process is called completion. When applied to the integers with the ideal $p \mathbb{Z}$, one gets the ring of $p$-adic integers, and more generally applied to rings of integers in number fields one can get rings of integers in local fields.

Again, one of the things that is nice about completion is that one can in fact do analysis in a reasonable way. A standard example is given by Hensel's lemma, which allows you to lift approximate solutions to exact solutions (in the completion). The completions turn out to provide information about the integers themselves (the starting point): for instance, it is a general principle that if something is true over $\mathbb{R}$ and all the $p$-adic fields $\mathbb{Q}_p$, then it should be true over $\mathbb{Q}$. The usual example is the Hasse-Minkowski theorem for quadratic forms, but the analog for other polynomials is false.

As for the infinite Galois group: the answer is that it is an inverse limit of the Galois groups of finite subextensions, and as such inherits a topology in a similar way as the completion.

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dear Akhil,thanks a lot for the answer,but can you be more specific by your words"Without the completeness property of the real numbers, most of ordinary mathematical analysis would fail: try doing integration over the rationals, for instance. " and more ,what is doing anlysis in various field? –  Yoshinobu Osawa Feb 1 '12 at 4:20
    
@Yoshinobu: Most of the basic theorems of analysis (e.g. things like the intermediate value theorem) rely intrinsically on the completeness of the real numbers; this is what guarantees that $\mathbb{R}$ is a well-behaved topological space (e.g. locally compact, connected -- these are the properties that underlie many of these theorems). As for your latter question, you may wish to look at $p$-adic analysis. –  Akhil Mathew Feb 1 '12 at 6:26

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