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

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- Mar 12th 2006, 04:19 PMThePerfectHackerIsomorphic Fields
If $\displaystyle F\leq E$ and $\displaystyle F\leq K$ are algebraically closed fields extension of $\displaystyle F$, is it true that $\displaystyle E\simeq K$?

- Mar 13th 2006, 12:51 PMrgep
No. For example, the field

of all algebraic numbers is algebraically closed, as is the field*A*of complex numbers, and both are extensions of the rationals*C*. But*Q*is countable and*A*is not: so they cannot be isomorphic.*C* - Mar 13th 2006, 01:22 PMThePerfectHackerQuote:

Originally Posted by**rgep**

- Mar 13th 2006, 09:50 PMrgep
Well, I wasn't the first to prove any of those facts!