I was thinking about this and was not able to prove this. I believe that if this is true then it might be useful in field theory.

If, $\displaystyle F\leq E$ are fields, and $\displaystyle \alpha,\beta\in E$ algebraic over $\displaystyle F$ and $\displaystyle [F(\alpha):F]=[F(\beta):F]$.

Prove (or disprove):

$\displaystyle F(\alpha)=F(\beta)$.

???