Field extension of composite degree has a non-trivial sub-extension Let $E/F$ be an extension of fields with $[E:F]$ composite (not prime). Must there be a field $L$ contained between $E$ and $F$ which is not equal to either $E$ or $F$? To prove this is true, it suffices to produce an element $x\in E$ such that $F(x) \not= E$, but I cannot find a way to produce such an element. Any ideas?

I don't know how you can find a primitive element abstractly but, for a particular extension with the fields explicitly stated, it must be useful to use the proof of the Theorem of Steinitz on finite extensions. It gives you a method to find a primitive element in the form 
$$\alpha + \lambda \beta,\quad \alpha, \beta\in E,\, \lambda \in F.$$
You can find the proof of this theorem in

Bastida, Julio. "Field Extensions and Galois Tehory". Page 155

