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Let $K/F$ be a finite extension with $K$ algebraically closed.

How can I show that $\mathrm{char}(F)=0$ and $K=F((-1)^{1/2})$ ?

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    This is the [Artin-Schreier Theorem](http://www.math.uconn.edu/~kconrad/blurbs/galoistheory/artinschreier.pdf). The link is to an exposition of it by [Keith Conrad](http://math.stackexchange.com/users/619/kcd).2012-04-18

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You could show that $F$ is real closed, see http://en.wikipedia.org/wiki/Real_closed_field for various equivalent definitions. $\operatorname{char}(F) = 0$ follows from the fact that the field is ordered, and $K = F(\sqrt{-1})$ is one of the equivalent definitions.

Edit: As marlu points out, you need $F \neq K$.