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I'm currently reading about local and global fields in number theory. I have trouble seeing the point or exactly how they help answer questions about e.g. number fields. To be more specific:

  1. What specifically gets simplified when we move to the local case?

  2. Is there some specific "theme" when using local methods? To be more specific: What type of results can usually be reduced to the local case? What type of results generally can't?

  3. What are the standard methods for lifting local results to global?

My current problem seems to be that I can't see the forest for the trees.

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You should check out the comments and answers to the recent MO post concerning the relevance of the real numbers to number theory. Most of them cover at least parts of your question more generally. –  Ramsey Apr 14 '11 at 5:21
    
Diophantine equations (and more generally, first-order problems) become decidable when you move to the local case. See en.wikipedia.org/wiki/Hilbert's_tenth_problem for example. –  S. Carnahan Apr 14 '11 at 8:13
    
See the end of Washington's book on cyclotomic fields, where he uses algebraic number theory to show how the Kronecker-Weber theorem for Q would follow from the K-W theorem from every Q_p and then he proceeds to use special features of p-adic fields to prove the K-W theorem for every Q_p. –  KConrad Apr 18 '11 at 5:19

2 Answers 2

  1. The completion of a number field at a prime $p$ is a principal ideal domain with a single maximal ideal. The unit group has a relatively simple structure. Thus two of the main culprits for making life difficult in number fields disappear locally. In addition, the difficult part of the Galois group, namely the decomposition group, also disappears in the local case.

  2. Losely speaking, questions concerning a single prime ideal (inertia group, ramification groups, \ldots) have a good chance of making sense locally. Global questions, like the quadratic reciprocity law, don't, at least not in the usual formulation.

  3. The simplest tool is Hensel's Lemma. Other than that I would not speak of a method for "lifting" results from local to global. What you do is compare the global result with the collection of all local results, and this is highly specific to the problem you're looking at. In some cases, like the Kronecker-Weber theorem (see Cassels' book on local fields), the problems are easily overcome, in others (embedding problems in Galois theory) they're not.

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To complement this very nice answer (which I would have up-ed twice ;-) ) : you could have a look at wikipedia's page on the Hasse principle. –  Julien Puydt Apr 14 '11 at 10:09
    
Why do you write that the decomposition group disappears in the local case? In that case the decomposition group from the global Galois extension is the local Galois group. –  KConrad Apr 18 '11 at 5:18
    
Typical case of Galois confusion: I meant that the decomposition field "disappears": there is no splitting of prime ideals, just inertia and ramification. –  Franz Lemmermeyer Apr 18 '11 at 8:32

To start with part 3: "local-global principles" of various kinds are one of the big themes in number theory, at least. Starting with, for example, a positive integer k being a square if and only if it is a square modulo all primes. That doesn't explicitly use local fields; but extensions to the idea, going under the general name of "Hasse principle", do use local fields in their formulation. A major effort in Diophantine equations, for the theory of existence of rational solutions, has been to understand when the Hasse principle holds; and when it doesn't to explain how to modify it.

To answer 1 with an example: class field theory is much easier in the local case, and gives a relatively slick theory. Again the approach is associated with the name of Hasse. When you move to "non-abelian class field theory", a.k.a. the Langlands philosophy, local fields are part of the basic formulation (adelic).

My attitude to 2 is that we don't really know the scope, and that is part of the jury being out on "number theory". There are "p-adic analogues" of many things. My advisor used to say that it was sheer prejudice and force of habit that the real numbers were the first local field taught. That was a joke, but there is a grain of truth in it. It is probably helpful to see the "empire building" of local fields as coming out of classical techniques that work for quadratic forms and cyclotomic fields (Kummer) and have contributed to many other areas by now, in different ways (e.g. non-archimedean Lie groups).

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