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

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 .
