Suppose that $L:K$ is an algebraic and separable extension, and that for every $\alpha \in L$, the minimal polynomial of $\alpha$ over $K$ has degree at most $n$. Then $[L:K]\leq n$.
I'm having trouble proving this. The material that I am studying uses the following definition of separable:
Separable: an irreducible polynomial $f \in K[t]$ is separable over $K$ if is has no repeated roots in the algebraic closure of K. An algebraic extension $L:K$ is separable if every $\alpha \in L$ is algebraic over $K$ and the minimal polynomial of $\alpha$ over $K$ is separable over $K$.
I think this is to do with the primitive element theorem or perhaps the Tower Law, but I can't puzzle it out. Any help appreciated.