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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 $\mathbb{R} \cap \overline{\mathbb{Q}}$.

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    $\begingroup$ What is the artin-schreier? I think it is about characteristic p. What is the statement? $\endgroup$
    – nicksohn
    Jan 2, 2016 at 8:50
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    $\begingroup$ @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 $i^2 = -1$. $\endgroup$ Jan 2, 2016 at 9:19
  • $\begingroup$ May I ask you the source? I want to study this theorem, statement, PROOF, and more examples... I can't find related articles.. Thank you! $\endgroup$
    – nicksohn
    Jan 2, 2016 at 9:37
  • $\begingroup$ @nicksohn: this is Theorem 171 in Pete Clark's notes on field theory: math.uga.edu/~pete/FieldTheory.pdf $\endgroup$ Jan 2, 2016 at 9:40
  • $\begingroup$ Thanks! Just to be sure, In our case, Because (iv) implies (iii), Right? $\endgroup$
    – nicksohn
    Jan 2, 2016 at 9:53

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