# Finiteness of the Algebraic Closure

Let $\mathbb R$ be the field of real numbers. Its algebraic closure, the field $\mathbb C$, is a finite extension of $\mathbb R$, which has degree 2.

Are there other examples of fields (not algebraic closed) such that its algebraic closure is a finite extension?

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The Artin-Schreier theorem asserts that these are precisely the real closed fields, which roughly speaking are the fields which behave like $\mathbb{R}$, and that their algebraic closures have degree $2$ and are given by adjoining a square root of $-1$. The Wikipedia article gives several examples; the simplest one is probably the real algebraic numbers $\mathbb{R} \cap \overline{\mathbb{Q}}$.
@nicksohn: "Artin-Schreier" is attached to two different things which are unrelated. The Artin-Schreier theorem asserts that $k$ is a field whose algebraic closure is a nontrivial finite extension of $k$ iff $k$ is a real closed field; in this case the algebraic closure is $k[i]$ where $i^2 = -1$. – Qiaochu Yuan Jan 2 at 9:19