Algebraic Number Theory and the Kronecker-Weber Theorem

The goal of this work is to prove the Kronecker-Weber theorem, an important first step to classifying abelian extensions of number fields. In chapter 1, we review the crucial concepts of Dedekind rings and ramification. Chapter 2 proceeds to study cyclotomic fields, ultimately developing the tools of ramification groups and the different. In chapter 3 we prove the main theorem, including two different proofs for the key statement to which we reduce the theorem for odd primes. We conclude with a brief look at the next steps, namely class field theory and Kronecker's Jugendtraum.