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Note finally that this answer does not depend on the fact that $K$ is the cyclotomic field of level $n$, or even the fact that the base field is $\Q$. It applies to any galoisian tower $E|K|F$: the associated short exact sequence $$1\to\Gal(E|K)\to\Gal(E|F)\to\Gal(K|F)\to1$$ splits if the degrees $[E:K]$, $[K:F]$ are mutually prime.