$L_1L_2/K$ is separable. justify it

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.

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

• Every finite extension of $\Bbb Q$ is separable. – Lord Shark the Unknown Feb 26 at 18:56
• Your answer doesn't work, since $[L_1L_2:K] = [L_1:K] [L_2:K]$ is not true in general. – darij grinberg Feb 26 at 18:56
• Also, how do you define "separable"? (This is the first question that should be asked from everyone who asks anything about separable field extensions. There are several definitions, and proving their equivalence is no easier than solving most exercises about separable extensions.) – darij grinberg Feb 26 at 18:57