I have the following exercise:
Let $E / \mathbb{Q}$ be a finite dimensional field extension of $\mathbb{Q}$.
If $[E :\mathbb{Q} ]=6$ and $\mathbb{Q} \subset F_{1} \subset E, \quad \mathbb{Q} \subset F_{2} \subset E$ are subextensions with
$[F_{1}:\mathbb{Q}]=2, \quad [F_{2}:\mathbb{Q}]=3$.
Let $z_{1} \in F_{1}$ and $z_{2} \in F_{2}$ with $z_{1}, z_{2} \notin \mathbb{Q}$. Must there exist $\lambda_{1}, \lambda_{2} \in \mathbb{Q}$ such that
$\mathbb{Q}\left(\lambda_{1} z_{1}+\lambda_{2} z_{2}\right)=E$.
What happens if we assume $[E :\mathbb{Q} ]=12$, $[F_{1}:\mathbb{Q}]=4$, $[F_{2}:\mathbb{Q}]=3$?
My attempt to answer was to mimic the proof of the primitive element theorem, where the fact that the base field is infinite is used, but I cannot understand how I should use the given degrees of the extensions.
Thank you in advance for the help.