# Finite extensions of fields that are algebraically closed

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.

• @user You should perhaps turn that into an answer... – Bruno Joyal Nov 27 '13 at 23:08