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 $\text{Gal}(K^{\rm ab}/H)$, for $H$ the Hilbert class field, isomorphic to $\widehat{\mathcal{O}}_K^{\times}$ modulo the closure of the global units?maximal abelian extension of (this would be in conflict with$K$ isomorphic, via the first questionArtin map, so I am also implicitly asking how to tweak this statement). Can it be at least written as athe quotient of $\widehat{\mathcal{O}}_K^{\times}$ modulo the closure of some subgroup (maybeby the closure of $\mathcal{O}_K^{\times,+}$$\mathcal{O}_{K,+}^{\times}$, the group of totally positive units? It would work for
- Question 3 The kernel of the Artin map $K=\mathbf{Q}$ and imaginary quadratic fields at least) related to$$\Phi_{H/K}:\mathbf{A}_K^{\times}\to\text{Gal}(H/K)$$
is $K$ and/or$K^{\times}\cdot((K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}\times \widehat{\mathcal{O}}_K^{\times})$. Is $H$$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?
As a side question, I know that when $K$ has class number one, the idéle group is $\mathbf{A}_K^{\times} = (K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}\times K^{\times}\times\widehat{\mathcal{O}}_K^{\times}$. Is there a description of $\mathbf{A}_K^{\times}$ in general, involving $H$?
Remark. On the side question, maybe $\mathbf{A}_K^{\times}$ can be realized as an extension of topological groups, involving the direct product $(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}\times K^{\times}\times\widehat{\mathcal{O}}_K^{\times}$ as the first of the terms? My hope is that one can have, say, $(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}\times K^{\times}\times\widehat{\mathcal{O}}_K^{\times}$think I see how to be the image under the norm mapdo $$\text{Norm}_{H/K}: \mathbf{A}_H^{\times}\to \mathbf{A}_K^{\times}$$(3), and so $\mathbf{A}_K^{\times}$ can be realized as an extension of the class group by the direct product $(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}\times K^{\times}\times\widehat{\mathcal{O}}_K^{\times}$. I don't know if this is trueroughly get (is it as a consequence of the Principal Ideal Theorem?1) and (2), but a statement like this is whatthough I'd be interested in! If true, finding a reference would be great, if any.
