Talk:Real closed field

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A real closed field is an ordered field F in which any of the following identical conditions are true:

(1) Every non-negative element of F has a square root in F, and any polynomial of odd degree has at least one root in F.

(2) The field extension F(\sqrt{-1}) is algebraically closed.

(3) F has no proper algebraic extension to an ordered field.

If F is any ordered field, the Artin-Schreier theorem states that F has an algebraic extension, the real closure K of F, such that K is real closed and whose ordering is an extension of the ordering on F. For example, the real closure of the rational numbers are the real algebraic numbers.

This article desperately needs to be reviewed by someone familiar with the subject. I just removed some obvious nonsense, but there are still some highly suspicious statements (e.g., the definition of superreal field does not look to me like something which could guarantee real-closedness). EJ 22:18, 6 May 2005 (UTC)

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