Let $f(x)\in \mathbb{Q} [x]$ irreducible polynomial of degree 4, $u \in \mathbb{C}$ a root of $f(x)$. Prove that there are not subfields $K$ such that $\mathbb{Q} \subset K \subset \mathbb{Q} (u)$ if and only if the galois group of $f(x)$, $G(N/\mathbb{Q}) \cong G_f^\theta$, is $A_4$ or $S_4$.
$N$ is the splitting field of $f(x)$ over $\mathbb{Q}$ and $K$ is a proper subfield between $\mathbb{Q}$ and $\mathbb{Q} (u)$
How to prove it?