a question about field extensions and tower formula

Question

if $K|F$ is a field extension & $a_1,a_2,...,a_n$ are the elements of $K$ which are algebraic on $F$ , we know that $[F(a_1,a_2,...,a_n):F]=<\Pi_{i=1}^n[F(a_i):F]$,it can be proved by induction on $n$.

is it also true that $ (\Pi_{i=1}^n[F(a_i):F])$ is divisible by $[F(a_1,a_2,...,a_n):F]$? is any condition necessary for this?

any proof or counter example is welcomed.

thank u very much

Answer 1

Hint: Consider the case $F=\Bbb{Q}$, $a_1=\root3\of2$, $a_2=\omega\root3\of2$, where $\omega=(-1+i\sqrt3)/2$ is a primitive third root of unity.