The Kronecker Weber Theorem and Concepts in Algebraic Number Theory

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.