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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.

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    $\begingroup$ I don't think that 1) is true of the class group of $K$ is nontrivial, and similarly for 2) if the narrow class group is nontrivial. $\endgroup$
    – Aurel
    Commented Oct 26, 2023 at 16:50
  • $\begingroup$ @Aurel I see. The right statement should be, for (1), the Galois group is isomorphic to the quotient of $\mathbf{A}_K^{\times}$ by the closure of $K^{\times}\cdot(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times, 0}$ (where the superscript $0$ means "connected component of the identity"), and for (2) it should be the same but with $(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times, 0}$ replaced by all of $(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}$. Am I right? $\endgroup$
    – Tim
    Commented Oct 26, 2023 at 17:33
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    $\begingroup$ This is the correct statement for (2). For (1) your formulation is correct if $K$ itself is totally real; but if $K$ isn't totally real, "the maximal totally real subfield of $K^{\mathrm{ab}}$" will not contain $K$! You want to replace "totally real" with "unramified at the real places of $K$" (i.e. as close to totally real as $K$ itself is). $\endgroup$
    – David Loeffler
    Commented Nov 6, 2023 at 20:29

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