Let $K$ be an extension field of $F$ and $L$ an extension field of $K$. If $a \in L$ is algebraic over $F$, show that $[K(a) : K] \leq [F(a) : F]$.
My idea: If $m(x)$ is the minimal polynomial of $a$ over $F$, I know that this polynomial is monic and irreducible. Also, I see that $F(a)$ is a subfield of $K(a)$. I think that this fact could help to solve this exercise. Since $m(x)$ lies also in $K(a)$. So if $K(a)$ is irreducible in $K(a)$, we get the equality, otherwise $[K(a) : K] < [F(a) : F]$. Am I right?
This is my first encounter with field theory and I don't see the definitions and theorems very clearly yet. Any help would be greatly appreciated.