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This is a big list of applications of the class number formula and its generalizations. I'll start:

  1. The solution to Gauss's class number problem for imaginary quadratic fields, and more generally the Brauer-Siegel theorem.
  2. The comparison of the class number of a field with the class numbers of its subfields
  3. The application of the Beilinson-Bloch conjecture to the arithmetic Bogolomov-Miyoaka-Yau inequality. See here.
  4. The explicit construction of class fields of totally real fields via Stark's conjectures.

What else?

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Perhaps it would be better if each entry in the list came with some more substantial exposition than just a link. You might do this by putting them in separate answers, and writing a short summary of what you're linking. This way, your question becomes a place to actually read about the class number formula, instead of just a list of links. Secondly, the lack of an explanation of the class number formula or a link to say a Wikipedia page suggests you only intend the list for people who already know the formula well, and I think the question would be more attractive if you changed that. – S. Carnahan Nov 24 '12 at 6:27
Thanks for your suggestions. I like them. I'll implement them when I have a chance (though it may take me some time to do so, as there's a lot of expository material to write up). – Jonah Sinick Nov 24 '12 at 6:57
In the same spirit of Scott's comment I suggest to give a link to the class number formula itself and to introduce it in a few sentences in the body of the question. – Gil Kalai Nov 24 '12 at 19:33

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