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Let $\mathbb R$ be the field of real numbers. Its algebraic 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 $\mathbb{R} \cap \overline{\mathbb{Q}}$.

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