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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.

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Why don't you tell us which books you are reading at on class field theory and say some more about your background in number theory, so people can give answers tailored to what you know and don't know? If you're not sure where class field theory came from, take a look at – KConrad Apr 18 '11 at 4:40
I've read Neukirch ANT Ch I-II, Lang ANT up to CFT chapters. I also have Janusz book. – dstt Apr 18 '11 at 4:48
The book "Class Field Theory: From Theory to Practice" by George Gras should have many examples. But also it's worth keeping in mind the following comment of Miles Reid from his Undergraduate Algebraic Geometry (p. 117): "When general theory proves the existence of some construction, then doing it [explicitly] is a useful exercise that helps one to keep a grip on reality, [but] this should not however be allowed to obscure the fact that the theory is really designed to handle the complicated cases, when explicit computations will often not tell us anything." – KConrad Apr 18 '11 at 5:15
I am planning a book called "Exercises in class field theory", which I will start working on this summer. If you send me an email I can keep you updated. – Franz Lemmermeyer Apr 18 '11 at 8:34
I suspect that professors are unlikely to ask you to compute a class field on a qual, outside the cases covered by Kronecker-Weber and perhaps by torsion points on elliptic curves, precisely because these computations are so difficult. If I were on a qual committee with a CFT syllabus, I might ask things like: "Show that, if $p$ does not divide the class number of $\mathbb{Q}(\zeta_p)$, and $u$ is a unit of $\mathbb{Q}(\zeta_p)$ which is $1 \mod p \mathbb{Z}(\zeta_p)$, then $u$ is a $p$-th power in $\mathbb{Q}(\zeta_p)$." You should find out what some older students have been asked. – David Speyer Apr 18 '11 at 13:59

You should take a look at Cox's wonderful book "Primes of the form $x^2+n y^2$".

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Thanks, this book looks great! It doesn't do local class field theory though. – dstt Apr 18 '11 at 4:38
@dstt: Are you looking for applications of CFT or are you looking for computational exercises to help you prepare for your quals? These are two very different questions. – Timothy Chow Apr 18 '11 at 14:18
great book, great application on how to use CFT! – Tommaso Centeleghe Apr 18 '11 at 21:18
@TChow: It's both really. – dstt Apr 21 '11 at 3:17
This book truly is great. Just checked it out yesterday and wish I had a year ago. Shame that it currently costs over $100 on Amazon though. – Jon Yard May 6 '11 at 16:04

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.

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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).

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Not the "London proceedings" but the Proceedings of the instructional conference held at the University of Sussex, Brighton, September 1–17, 1965. – Chandan Singh Dalawat Apr 19 '11 at 5:55

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.

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You might like the answers to this question: Image of norm map for local field

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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:

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