Let $f \in \mathbb{Q}[t]$ be an irreducible polynomial of degree $3$ and $L_f \subset \mathbb{C}$ a splitting field of $f$ over $\mathbb{Q}$. Prove that there is no field $E$ with $\mathbb{Q} \subset E \subset L_f$ such that the extension $L_f \vert E$ has degree $2$, or there are exactly three fields satisfying those conditions.
I'm really stuck on this problem. I know that somewhere I should use the Fundamental Theory of Galois theory, but I don't know how.
Furtheremore, I know that since $f$ is an irreducible polynomial of degree $3$, its Galois group must be $\mathbb{Z}_3$ or $S_3$. If it is the last one, I know that $S_3$ has exactly three subgroups of order two, $(1,2)$, $(1,3)$ and $(2,3)$, and if it is $\mathbb{Z}_3$, there are no subgroups of order $2$, but I don't know how this two facts are related with the number of fields of degre $2$. Any help with this willbe highly appreciate, and thanks in advance!