1. ## Isomorphic Fields

If $F\leq E$ and $F\leq K$ are algebraically closed fields extension of $F$, is it true that $E\simeq K$?

2. 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.

3. Originally Posted by rgep
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.
Nice (dis)proof I like it, is it yours?

4. Well, I wasn't the first to prove any of those facts!