Real-closed field
In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers.
A real closed field is a field F in which any of the following equivalent conditions are true:
If F is an ordered field, the Artin–Schreier theorem states that F has an algebraic extension, called the real closure K of F, such that K is a real closed field whose ordering is an extension of the given ordering on F, and is unique up to a unique isomorphism of fields identical on F (note that every ring homomorphism between real closed fields automatically is order preserving, because x ≤ y if and only if ∃z y = x + z2). For example, the real closure of the ordered field of rational numbers is the field of real algebraic numbers. The theorem is named for Emil Artin and Otto Schreier, who proved it in 1926.
If (F,P) is an ordered field, and E is a Galois extension of F, then by Zorn's Lemma there is a maximal ordered field extension (M,Q) with M a subfield of E containing F and the order on M extending P. M, together with its ordering Q, is called the relative real closure of (F,P) in E. We call (F,P) real closed relative to E if M is just F. When E is the algebraic closure of F the relative real closure of F in E is actually the real closure of F described earlier.
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