An Elementary Proof of the Local Kronecker-Weber Theorem

Abstract

Let $K$ be a local field. Lubin and Tate have shown how to explicitly construct an abelian extension of $K$ which they prove to be the maximal abelian extension. Their proof of this result uses local class field theory. When $K$ is a $p$-adic field we give an elementary proof which even avoids the use of higher ramification groups. Instead we rely on facts about the principal units in a finite abelian extension of $K$ as a module for the Galois group.