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?

In a previous answer, I considered the splitting field $E$ of
$$ f(x) = (x^3 + x + 1)^2 + 1 $$
Doing calculation with sage, I determined the Galois group $\text{Gal}(E/F)$ has 72 elements. It has subgroups of 8 elements, but no subgroups of 24 elements.
Therefore, $E/\mathbb{Q}$ contains a degree 9 subextension $K/\mathbb{Q}$, and $K/\mathbb{Q}$ does not contain any degree 3 subextensions.
