Let 'a' belong to C and be algebraic over Q, suppose F contained in C is a subfield. Show [F(a) : F] <= [Q(a): Q]

Question

I know that since $a$ is algebraic over $Q$, this means that $Q(a)$ is a finite extension of $Q$ so $[Q(a) : Q] \leq n$

so we can definite a basis $\{v_1, ..........., v_n\}$ for $Q(a)$

Im stuck on how to proceed. Any help would be much appreciated

Answer 1

Hint: Is $\Bbb Q$ a subset of $F$?

More hints: Note that $1$ is in $F$. So $1+1$ is in $F$, $1+1+\ldots+1$ is in $F$. Also, $\frac{1+1}{1+1+1+1+1}$ must be in $F$...