This is a big list of applications of the class number formula and its generalizations. I'll start:
- The solution to Gauss's class number problem for imaginary quadratic fields, and more generally the Brauer-Siegel theorem.
- The comparison of the class number of a field with the class numbers of its subfields.
- The application of the Beilinson-Bloch conjecture to the arithmetic Bogomolov-Miyoaka-Yau inequality. See here.
- The explicit construction of class fields of totally real fields via Stark's conjectures.
What else?
