Examples of Using Class Field Theory I'm trying to learn class field theory and I'm wondering if anyone knows of any good sources with a bunch of examples on how to actually use it? This can be anything from books to course notes to course websites with solved homework. Interesting examples would be something like constructing specific extensions of $\mathbb{Q}$ and $\mathbb{Q}_p$, determining splittings of primes in more complicated extensions than the quadratics or anything else that is "concrete" where it might be useful.
My problems seems to be that while I can understand the actual statements it still seems like I can't see how to actually use it for any practical computations. By looking at the definitions it just seems like most objects aren't terribly computable. Most books just seem to have just some fairly trivial examples like e.g. finding the Hilbert Class Field of something like $\mathbb{Q}(\sqrt{-5})$ and the examples exists to just give an example of some defined object and do not actually use the theorems for anything.
 A: As I have written in your question on SE, if you want to know how to actually compute polynomials that give you ring class fields for a given modulus, then Cohen's Advanced Topics in Computational Number Theory is a very good resource. For a "real life example", you can have a look at section 3.1 of this paper, where I spell out how to find dihedral extensions of $\mathbb{Q}$ with a given intermediate quadratic (again, if you want polynomials generating these extensions, then see Cohen). See also the articles by Yui with various collaborators, many of which use and make explicit the constructions of class field theory.
A: In the London proceedings (Cassels-Froehlich), Tate and Serre have written some (classical) exercises regarding CFT (i.e. deducing higher reciprocity laws from Artin's reciprocity law, the Hasse-Minkowski theorem and a few others I can't recall right now).
A: I suggest taking a look also at the two books "A classical invitation to algebraic numbers and class fields" and "Introduction to the construction of class fields" by Harvey Cohn. While dealing only with global class field theory, they adopt a very concrete approach (much in the same spirit as the book of Cox in Some guy's answer) and (if memory serves me well) offer explicit examples of construction of class fields which are quite hard to find elsewhere.
A: You might like the answers to this question: Image of norm map for local field 
A: You should take a look at Cox's wonderful book "Primes of the form $x^2+n y^2$".
A: You can prove that the class number of a cyclotomic number field of an odd prime order is divisible by that of its subfield by using class field theory.
Since it has the unique quadratic subfield and its class number can be relatively easily computed when the discriminant is small, you can get useful information of the class number of the cyclotomic number field.
For example, you can find the proof here:
https://math.stackexchange.com/questions/175718/on-the-class-number-of-a-cyclotomic-number-field-of-an-odd-prime-order
