While studying field theory, I encountered the following question, which is expected to be true, but hard to prove for me.
Claim:
Let $F$ be a field with ${\rm char}F = 0$ and let $F \left(\beta_1\right), F\left(\beta_1, \beta_2\right)$ be algebraic extensions of $F$. Then let $E$ be the splitting field of a polynomial $f(x)\in F[x]$ over $F$. The claim is that if $\beta_2 \in E\left(\beta_1\right)$, we also have $$\beta_2 \in \left(F\left(\beta_1, \beta_2\right)\cap E \right)\left(\beta_1\right).$$
I thought this is plausible because if $\beta_2$ is expressed by elements of $E$ and $\beta_1$, it should also be expressed using only elements of $E$ which are also expressed by $\beta_1$, $\beta_2$ together with $\beta_1$. For example, if $F=\mathbb{Q}$ and $\beta_1 = \sqrt{3}$, $\beta_2 = \sqrt{6}$ and $E = \mathbb{Q}\left(\sqrt2\right)$, it follows $F\left(\beta_1,\beta_2\right)\cap E = E$ and the claim holds obviously.
So I tried to prove or disprove it by investigating some more examples but failed. Could someone help me with this?