I have read that the main goal of Class Field Theory is to characterize all abelian extensions of a number field $K$ in terms of parameters of $K$ only.
We know that for $K=\mathbb{Q}$, the Kronecker-Weber Theorem says that any abelian extension is contained inside some cyclotomic extension.
We know that for an abelian extension $L|K$, the Artin map is surjective.
What I want to know is this:
Let $K=\mathbb{Q}$. Are we getting any extra information about the abelian extensions of $K$ (which we cannot deduce from the Kronecker-Weber Theorem) from the fact that the artin map is surjective?