Let $K/F$ be a finite extension with $K$ algebraically closed.
How can I show that $\mathrm{char}(F)=0$ and $K=F((-1)^{1/2})$ ?
Let $K/F$ be a finite extension with $K$ algebraically closed.
How can I show that $\mathrm{char}(F)=0$ and $K=F((-1)^{1/2})$ ?
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 $F \neq K$.