In this bachelor's thesis we give a complete proof of the Kronecker-Weber theorem, which states that every finite abelian extension over the field of rational numbers Q is contained in some cyclotomic extension of Q. We prove it using a local-global principle: the K-W theorem is true for Q if and only if its is true for Qp, the field of p-adic numbers, for every prime p; and then we prove the local case. In order to show this local-global principle, we study the p-adic fields Qp and their finite extensions, and characterize them as completions of number fields.