I'm struggling with this problem. Given $E/F$, $E_1/F$, and $E_2/F$ are field extensions such that $E_1, E_2 \leq E$ and $E_1/F$ and $E_2/F$ are finite, we want to show that $[E_1E_2:E_1] = [E_2 : E_1 \cap E_2]$. Any hints would be helpful.
2026-04-04 04:24:16.1775276656
Showing $[E_1E_2 : E_1] = [E_2 : E_1 \cap E_2]$.
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Counter example:
$$E_1=\mathbb{Q}(\sqrt[3]{2})$$ $$E_2=\mathbb{Q}(\sqrt[3]{2}\zeta_3)$$