On the structure of the maximal abelian Galois group of a number field

Question

Let $K$ be a number field. I am wondering if the following exact sequence $$1 \longrightarrow[\widehat{\mathcal O}_K^\times] \longrightarrow Gal(K^{ab}/K) \overset{\pi}{\longrightarrow} Cl_K \longrightarrow 1$$ splits, i.e. if there is a homomorphism $s:Cl_K \to Gal(K^{ab}/K)$ such that $s \circ \pi = id$. Here $[\widehat{\mathcal O}_K^\times]$ stands for the image of $\widehat{\mathcal O}_K^\times$ in $Gal(K^{ab}/K)$ under Artin's reciprocity morphism and $Cl_K$ denotes the ideal class group of $K$.

Equivalently we can rewrite the above sequence idelically as

$$1 \to \widehat{\mathcal O}_K^\times / \overline{\mathcal O _{K,+} ^\times} \to \widehat{\mathcal O}_K^\natural / \overline{\mathcal O _{K,+}^\natural} \to \widehat{\mathcal O}_K^\natural / (\widehat{\mathcal O}_K^\times \cdot \mathcal O _{K,+}^\natural) \to 1,$$ where $\mathcal O _K ^\natural = \mathcal O _K - \{ 0 \} $, $\widehat{\mathcal O}_K^\natural = \mathbb A _{K,f}^\times \cap \widehat{\mathcal O}_K$. Further, subscript $+$ denotes totally positive elements and over lined objects are meant to be closures in the idele topology. (I don't know if this is helpful.)

Answer 1

Let $K = {\mathbb Q}(\sqrt{-5})$ and let $P = (3+2\sqrt{-5})$ denote a prime ideal of norm $29$ in the ring of integers of $K$. There does not exist a quadraic extension of $K$ ramified exactly at $P$, but there is one over the Hilbert class field $K(i)$ of $K$. This means that the ray class group modulo $P$ does not split into the product of the ideal class group, which has order $2$, and another group. In particular, the maximal abelian extension unramified outside of $P$ does not split into a compositum of an unramified extension and a purely ramified extension, and neither does $K^{\rm{ab}}$.