Reference request - infinite ramification theory I want to study the ramification theory for an infinite Galois extension. Regarding infinite Galois theory, I can find some references, but I cannot find anything that satisfies me about the ramification theory.
I would like to have also a systematic approach to the subject. I find only the definition of an unramified extension, but what does it mean that a prime is totally ramified in an infinite Galois extension? What are the basic strategies to prove that a prime is totally ramified using its inertia group?
 A: A basic treatment of infinite Galois theory is not likely to include a discussion of ramification because the latter is a more specialized topic, belonging to algebraic number theory, or, if you like, algebraic geometry. The second appendix to Washington's book on cyclotomic fields gives a very basic introduction to ramification of primes in infinite (Galois) extensions of number fields, and I would say this is a reasonable starting point if you have never seen this material.
In answer to your specific question about totally ramified primes, if $L/K$ is an arbitrary (i.e. not necessarily finite) Galois extension of a number field $K$, then a nonarchimedean/finite prime $\mathfrak{p}$ of $K$ is totally ramified in $L$ if for one (equivalently any) prime $\mathfrak{P}$ of $L$ lying above $\mathfrak{p}$, the inertia subgroup $I(\mathfrak{P}/\mathfrak{p})$ of $\mathrm{Gal}(L/K)$ is equal to $\mathrm{Gal}(L/K)$. 
The natural context for a more refined study of ramification of this kind is local, meaning the setting of extensions of nonarchimedean local fields (e.g. $\mathbf{Q}_p$). Indeed, ramification in the global context is already studied one (based) prime at a time, and once one fixes such a prime, one can complete with respect to the prime in order to move to a local context which in which all the desired ramification information is available. The local context is the one in which a clean treatment of higher ramification groups may be carried out. There are many books covering ramification in extensions of complete (maybe even just Henselian) discretely valued fields, for example Serre's classic Local Fields. Defining higher ramification groups for infinite extensions of such fields requires some contortions with the so-called upper and lower numbering (explained in Serre's book), but it is extremely important, for example, in defining the conductor of a $p$-adic Galois representation, in understanding $p$-adic Lie extensions and the related notion of an arithmetically profinite (APF) extension, and, generally, in $p$-adic Hodge theory.
I do not know any strategies for proving that a prime is totally ramified using its inertia group beyond just making use of the definition, which is, as I said above, that the inertia group is equal to the entire Galois group.
Edit: Regarding my last comment, I guess it is worth mentioning that the a prime is totally ramified in an infinite extension if and only if it is totally ramified in every finite subextension. 
A: May I also suggest Moreno, Advanced Analytic Number Theory, Part I: Ramification Theoretic Methods (Contemporary Mathematics, Volume 15). 
