Finitely generated field extensions 
If $F=K(u_1,\ldots,u_n)$ is a finitely generated extension of $K$ and $M$ is an intermediate field, then $M$ is a finitely generated extension of $K$. 

I'm not exactly sure how to start this problem. Any help at all would be a tremendous help.

Firstly, I encourage you to simply try this with the tower rule ( $[F:K] = [F:M][M:K]$ ). Remember that you can consider the transcendent degree separately. However, if that is insufficient, I direct you to this answer over at MO.
The accepted answer covers this question very nicely, including the cases for transcendence.
