Real Closed Field - Definitions

Definitions

A real closed field is a field F in which any of the following equivalent conditions are true:

1. F is elementarily equivalent to the real numbers. In other words it has the same first-order properties as the reals: any sentence in the first-order language of fields is true in F if and only if it is true in the reals.
2. There is a total order on F making it an ordered field such that, in this ordering, every positive element of F is a square in F and any polynomial of odd degree with coefficients in F has at least one root in F.
3. F is a formally real field such that every polynomial of odd degree with coefficients in F has at least one root in F, and for every element a of F there is b in F such that a = b2 or a = −b2.
4. F is not algebraically closed but its algebraic closure is a finite extension.
5. F is not algebraically closed but the field extension is algebraically closed.
6. There is an ordering on F which does not extend to an ordering on any proper algebraic extension of F.
7. F is a formally real field such that no proper algebraic extension of F is formally real. (In other words, the field is maximal in an algebraic closure with respect to the property of being formally real.)
8. There is an ordering on F making it an ordered field such that, in this ordering, the intermediate value theorem holds for all polynomials over F.
9. F is a field and a real closed ring.

If F is an ordered field (not just orderable, but a definite ordering is fixed as part of the structure), 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 (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 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 is a field (so this time, no order is fixed, and it is even not necessary to assume that F is orderable) then F still has a real closure, which in general is not a field anymore, but a real closed ring. For example the real closure of the field is the ring (the two copies correspond to the two orderings of ). Whereas the real closure of the ordered subfield of is again the field .