TITLE: Local methods in algebraic number theory
QUESTION [5 upvotes]: 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.

REPLY [8 votes]: 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.