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Let $F\subset K$ be a Galois extension such that $G(K,F)\cong S_3$. Is it true that $K$ is the splitting field of an irreducible cubic polynomial over $F$?

I really couldn't think of a counter example for this. Hence I was going to prove the statement.
So as we have $G(K,F)\cong S_3$ by looking at the lattice structure we can determine that there are four distinct intermediate fields. Let them be $A,B,C,D$ where ,
$[K,A]=[K,B]=[K,C]=[D,F]=2$ and
$[A,F]=[B,F]=[C,F]=[K,D]=3$

Also I know that if the degree of the extension $G\subset H$ is two then $\exists h\in H$ such that $H=G(\sqrt h)$.
But after this I couldn't proceed any further.
Appreciate your help.

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  • $\begingroup$ Hint: A is generated by one element over F (why?). What is the orbit of this element under the action of G(K, F)? $\endgroup$ – sss89 Mar 26 at 7:16

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