# A result when $G(K,F)\cong S_3$

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.