Is there always an intermediate field between non-prime field extension?

Question

If we have $[E:F]=n$, where $n$ is not a prime number but is finite, can we like prime factorize $n=p_1p_2...p_r$, so that we have $[E:F]=[E:K_1][K_1:K_2]...[K_{r-1}:F]$ and each of the $[K_i:K_{i+1}]$ has a prime degree of extension? Can we always find such intermediate fields $K_1, K_2,...,K_{r-1}?$

Thank you very much.

Answer 1

In general the answer to your question is negative. For an example let us pick an irreducible quartic polynomial $p(x)$ over the rationals with Galois group $S_4$. If $\alpha$ is one of its roots, then $E=\Bbb{Q}(\alpha)$, $F=\Bbb{Q}$, then $[E:F]=4$, but there are no intermediate fields between $E$ and $F$. For $E$ is the fixed field of a copy of $S_3$, and there are no intermediate subgroups $H$ of order 12 such that $$S_3\le H\le S_4.$$

Answer 2

No, it's not always possible. If it were, then in particular, every Galois field extension would be solvable, which we know to be false. See http://en.wikipedia.org/wiki/Galois_theory#Solvable_groups_and_solution_by_radicals .