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I was referred to this question in a previous post. I was wondering if something is known about the stronger condition, i.e. when $G$ is cyclic.

Some heuristics:

  • Every dihedral group $G \cong \mathbb{Z}/n\mathbb{Z} \rtimes \mathbb{Z}/2 \mathbb{Z}$ can be realized as a Galois group, i.e. there exists some extension $L$ over $\mathbb{Q}$ such that $G \cong Gal(L/\mathbb{Q})$ (see here, page 2).
  • Let $L$ be the ring class field of an order $\mathcal{O}$ in an imaginary quadratic field $K$. Then $L$ is a Galois extension of $\mathbb{Q}$ and its Galois group can be written as a semidirect product $$\textrm{Gal}(L/\mathbb{Q}) \cong \textrm{Gal}(L/K) \rtimes \mathbb{Z}/2 \mathbb{Z} \cong CL(K) \rtimes \mathbb{Z}/2 \mathbb{Z}$$ where the last isomorphism holds by Artin reciprocity.

However... both statements cannot be combined in a straightforward manner. In the proof of the first statement $L$ is an extension of the field $K=\mathbb{Q}(i)$ with $\textrm{Gal}(L/K) \cong \mathbb{Z}/n\mathbb{Z}$.

But we cannot combine this with the second statement (and Artin reciprocity) to argue that $\textrm{Gal}(L/K) \cong CL(K)$. The reason is that $L$ is not the HCF of $K=\mathbb{Q}(i)$ (see here) and therefore the isomorphism does not hold.

Some questions:

  • Can we fix the above problem somehow, or does my attempt necessarily lead to a dead end?
  • More generally: what is known in the literature about my original question?

Thanks in advance

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