If $\displaystyle F\leq E$ and $\displaystyle F\leq K$ are algebraically closed fields extension of $\displaystyle F$, is it true that $\displaystyle E\simeq K$?
No. For example, the field A of all algebraic numbers is algebraically closed, as is the field C of complex numbers, and both are extensions of the rationals Q. But A is countable and C is not: so they cannot be isomorphic.
No. For example, the field A of all algebraic numbers is algebraically closed, as is the field C of complex numbers, and both are extensions of the rationals Q. But A is countable and C is not: so they cannot be isomorphic.