I'm looking for a reference for the following facts from global class field theory that I found without proofs. I will state them as questions, just in case I get the statement wrong. We fix $K$ a number field.
- Question 1. Is the quotient of $\widehat{\mathcal{O}}_K^{\times}$ (the units in the profinite completion of $\mathcal{O}_K$) by the closure of the unit group $\mathcal{O}_K^{\times}$ of $\mathcal{O}_K$ isomorphic, via the Artin map, to the Galois group of the maximal totally real sub-field of the maximal abelian extension $K^{\rm ab}$?
- Question 2. Is the Galois group of the maximal abelian extension of $K$ isomorphic, via the Artin map, to the quotient of $\widehat{\mathcal{O}}_K^{\times}$ by the closure of $\mathcal{O}_{K,+}^{\times}$, the group of totally positive units?
- Question 3 The kernel of the Artin map $$\Phi_{H/K}:\mathbf{A}_K^{\times}\to\text{Gal}(H/K)$$ is $K^{\times}\cdot((K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}\times \widehat{\mathcal{O}}_K^{\times})$. Is $K^{\times}\cdot((K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}\times \widehat{\mathcal{O}}_K^{\times})$ in turn an extension of $(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}\times K^{\times}\times \widehat{\mathcal{O}}_K^{\times}$ by the unit group?
I think I see how to do (3), and I roughly get (1) and (2), though I'd be interested in finding a reference, if any.
