I am considering:
Suppose the degree is $n$. Then there is a basis consisting of $n$ elements ${x_i}$. Hence all intermidiate fields must have a set of basis consistingof $m< n$ elements, each of which is a linear combination of the $x_i$. Then notice that it suffices to prove that there is a finite number of intermidiate fields of a fixed degree. So suppose that $m$ is fixed. By what we have shown, the intermediate fields correspond to $m*n$ matrices of rank $m$, up to $m*m$ invertible matrices.