Suppose $E_1/F$ and $E_2/F$ are finite field extensions. The degree of the composite field $E_1E_2$ over $F$ is less or equal to the product of the degree of $E_1$ over $F$ times the degree of $E_2$ over $F$, i.e. $$ [E_1E_2:F] \leq [E_1:F] [E_2:F]. $$ See for example: The degree of a field extension is smaller than the product of the degrees of field extensions of intermediate fields that generate the field.
I suspect that $[E_1E_2:F]$ divides the product $[E_1:F] [E_2:F]$. Do you have a suggestion on how to prove this? Note we have the towers $F\subset E_1\subset E_1E_2$ and $F\subset E_2\subset E_1E_2$ that might be useful.