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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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    You should add the assumption that $K \neq F$, otherwise $K = F = \overline{\mathbb F_p}$ is a counterexample with characteristic $p$.2012-04-18
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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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