TITLE: Application of Kronecker Weber Theorem
QUESTION [7 upvotes]: I know this may be a very naive question. Please forgive my naivety. What are some of the applications of the Kronecker-Weber Theorem?

REPLY [7 votes]: A theorem like that of Kronecker and Weber is not measured in terms of applications, it is measured in terms of insight and the potential to generate powerful generalizations. It has given rise to Kronecker's theory of complex multiplications and to one of Hilbert's 23 problems, and is a guiding theorem for classical class field theory. It has various proofs, each of which uses an important technique (Lagrange resolvents and Stickelberger, Hilbert's abelian crossings, decomposition and ramification groups, local fields and a lot more).