I want to prove that
Gal($\mathbb{Q}^{ab}/\mathbb{Q}$) $\rightarrow$ $\hat{\mathbb{Z}}^×$
is an isomorphism by using the statement of class field theory
Gal($L/K$)$^{ab}$ $\rightarrow$ $I_K/N_{L/K}I_L$
is the isomorphism
where $I_L$ is the idele class group with respect to L.
I think that $L=\mathbb{Q}^{ab}$ and $K = \mathbb{Q}$ and want to prove $I_K/N_{L/K}I_L$ $\cong$ $\hat{\mathbb{Z}}^×$.
$I_K$ = $\prod_{p\ is\ prime}^{'}\mathbb{Q}_p$ for $\prod^{'}$ is restricted product.
But I don't know how to prove from here.
Please tell me how to prove!
Thanks in advance.
