### Abstract:

The Kronecker-Weber Theorem states that all abelian extensions are sub elds of cy-
clotomic elds. This paper considers a proof based on foundational concepts in algebraic
number theory that was presented by Greenberg in the 1970s. These concepts include
rings of integers as Dedekind domains, nite elds and residue extensions, rami ed primes,
properties of cyclotomic extensions, the norm of an element and the discriminant of an
extension.
The proof shows that all abelian extensions are sub elds of cyclotomic extensions by
breaking the problem down to cases of prime power order. An argument is made that is
analagous to the Chinese Remainder Theorem that the Galois group of the compositum of
two eld extensions is direct product of the Galois group of each eld extension. The proof
breaks down further into two cases, odd primes and powers of 2. The result then relies on
theorems pertaining to the rami ed prime in any given extension. It can be shown that for
the odd primes only one prime not dividing the order is rami ed in an abelian extension or
the only rami ed prime in the extension divides the order. Then it can be shown that this
extesion is a sub eld of a cyclotomic extension. A inductive argument based on valuation
theory is used to prove the power of 2 case.