Intersection with composite field

Question

Let $L/K,\ E_i/K$ be field extensions$(i=1\dots r)$ and for all $i$,$\ L\cap E_i=K$.

Then $L\cap E=K$ ?(where $E=E_1\cdots E_r$)

Answer 1

I think there are easy counterexamples. Suppose that $K = \mathbb{Q}, L = \mathbb{Q}(\sqrt{6}), E_1 = \mathbb{Q}(\sqrt{2}), E_2 = \mathbb{Q}(\sqrt{3})$. Then

$$E_i\cap L = K$$

for $i=1,2$. On the other hand $E_1\cdot E_2$ is an extension of $\mathbb{Q}$ generated by $\sqrt{2}$ and $\sqrt{3}$. Hence $\sqrt{6}\in E_1\cdot E_2$. Thus

$$E_1\cdot E_2 \cap L = L$$