Is it true that-
If $L_1/K$ and $L_2/K$ are extensions contained in a field $F$ and both are separable then $L_1L_2/K$ is separable.
If not true then give me any counter example.
Answer:
In general $ [L_1L_2:K] \leq [L_1:K] [L_2:K]$ and $ [L_1L_2:K] = [L_1:K] [L_2:K]$ iff $L_1 \cap L_2=K$.
So I think the given statement does not hold in general. For example,
let $K=\mathbb{Q}, \ L_1=\mathbb{Q}(\sqrt[3]{2}), \ L_2=\mathbb{Q}(\omega \sqrt[3]{2})$.
Clearly $L_1$ and $L_2$ are separable extension of $\mathbb{Q}$.
But $[L_1L_2:\mathbb{Q}]=6 \neq [L_1: \mathbb{Q}][L_2:\mathbb{Q}]$ though $L_1 \cap L_2=\mathbb{Q}$.
Thus $L_1L_2/K$ is not separable.
Am I right?
Help me