Linear independence over the compositum of two fields

Question

Suppose $K, F \subseteq L$ are fields. Say $\alpha_1, \dots \alpha_n \in L$ are both $K$- and $F$-linearly independent. Are they $KF$-linearly independent, where the compositum is taken inside $L$? This is somehow motivated by trying to understand towers of (transcendental) separable extensions -- namely, if $K \subseteq F \subseteq L$ is a tower of (not necessarily algebraic) field extensions and $L$ is separable over $K$, is then $L$ separable over $F$?

Answer 1

Try $K=\Bbb{Q}(2^{1/3}),F=\Bbb{Q}(e^{2i\pi/3} 2^{1/3}),L=KF=\Bbb{Q}(2^{1/3},e^{2i\pi/3}),\alpha_1=1,\alpha_2 = e^{2i\pi/3}$

Answer 2

Except for the trivial case (where $K\subseteq F$ or $F\subseteq K$), this is not true: if $\alpha\in K\setminus F$ and $\beta\in F\setminus K$, then $(1,\alpha\beta)$ is $K$-linearly independent and $F$-linearly independent, but clearly not $KF$-linearly independent.