$F\subset L \subset F[a]$ fields, prove L is created by $f_a$

Question

Let $F\subset L \subset F[a]$ be fields, and $f_a(x)=\sum_{k=0}^n a_k x^k$ is the minimal polynomial of $a$ over L. prove that $L=F\left[a_0,...,a_n\right]$.

I've tried a few ways but I can't manage to solve this, I'm probably missing something obvious...

Thanks :)

Answer 1

Hint (and actually an answer spoiling all suspense) : induction on $n$.