Local methods in algebraic number theory 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:


*

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

*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?

*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.
 A: *

*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.    

*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. 

*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.
A: 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).
