Computing $\text{Gal}(\mathbb{Q}^{ab}/\mathbb{Q})$

Question

Algebraic class field theory tells us that $\text{Gal}(\mathbb{Q}^{ab}/\mathbb{Q})$ is isomorphic to the group of connected components of the quotient $\mathbb{Q}^{\times}\backslash \mathbb{A}_{\mathbb{Q}}^{\times}\cong \prod_p \mathbb{Z_p}^{\times}\times \mathbb{R}_{>0}$, where $\mathbb{A}_{\mathbb{Q}}$ is the ring of adèles of $\mathbb{Q}$.

It's then said that the group of connected components is given by $\prod_p \mathbb{Z_p}^{\times}$, how can I see this?

Thank you very much in advance!

Answer 1

Try to prove this more general topological fact: if $\Gamma$ is totally disconnected and $\Lambda$ connected, then the connected components of the product space $\Gamma\times\Lambda$ are the subsets $\{x\}\times\Lambda$ for $x\in\Gamma$.