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Consider a finite field extension $K/k$ with $K$ algebraically closed. The immediate example that comes to mind is $\mathbb{C}/\mathbb{R}$, of degree $2$. Are there any other examples? Can we construct them of any desired degree $n$?

Motivation: I've recently been studying the Weil restriction of an affine scheme, which requires a finite extension of fields, and because algebraic geometry is cleaner over an algebraically closed field, I'd like one of my fields to be algebraically closed. Since there are no non-trivial finite extensions of an algebraically closed field, I'm left to find a field that has a finite extension that is algebraically closed.

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    $\begingroup$ @user You should perhaps turn that into an answer... $\endgroup$ – Bruno Joyal Nov 27 '13 at 23:08
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See the Artin-Schreier Theorem.

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