Let $K_v$ a local and $F$ a finite abelian extension of it.
Two questions:
Could somebody explain how local class field theory
associates naturally a character of $K_v^{\times}$ to $F$.
And, if $r \in K_v$ what is "the order" of $r$ in this character
of $K_v^{\times}$ associated to $F$.
The question arose from terminology used in this MO answer (first sentence) by Will Sawin specializing to $K_v= \mathbb{Q}_p$ and $r=p$.