the number of intermediate fields in a simple field extension of degree $n$

suppose that $K|F$ is a simple field extension with degree $n$,prove that the number of intermediate fields is less or equal $2^{n-1}$.

i've done this:

assume $K=F(a)$ and $L$ is a intermediate field .consider $f(x)\in F[x]$ the minimal polynomial of $a$ over $F$ and $g(x)\in L[x]$ the minimal polynomial of $a$ over $L$.
we have $g|f$ ,i want to make a surjective correspondence between the irreducible polynomials that divides $f$ and the intermediate fields.

is it a good idea?

any hint is welcomed!