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?

I roughly understand how (1) and (2) should go, though I'd be interested in finding a reference (so as to also doublecheck my work), if any.