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Let $K$ be a Galois number field with cyclic Galois group. Let $L$ be an abelian Galois number field such that $K\subseteq L$. Suppose that there are no Galois subextensions $\mathbb Q\subseteq F \subseteq L$ such that $\text{Gal}(F/\mathbb Q)$ is cyclic and $K\subsetneq F$.

Is it true then that the exact sequence $1\to \text{Gal}(L/K)\to \text{Gal}(L/\mathbb Q)\to \text{Gal}(K/\mathbb Q)\to 1$ splits?

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