Ordered field

2020 Mathematics Subject Classification: Primary: 12J15 [MSN][ZBL]

A totally ordered ring which is a field. The classical example is the field of real numbers with the usual order. By contrast, the field of complex numbers cannot be made into an ordered field, because a field admits an order turning it into an ordered field if and only if $-1$ cannot be written as a sum of squares. A field for which $-1$ cannot be written as a finite sum of squares is called a formally real field. The field of real numbers is a model of a formally real field. More generally, every ordered field is formally real.

An extension $P$ of an ordered field $k$ is said to be ordered if $P$ is an ordered field containing $k$ as ordered subfield. This takes place precisely when $-1$ cannot be written as a sum of elements of the form $\lambda x^2$, where $\lambda\in k$, $\lambda\geq0$ and $x\in P$. An ordered field is said to be real-closed if it contains no proper ordered algebraic extension. An order on a real-closed field is uniquely determined. The following conditions on an ordered field $k$ are equivalent: 1) $k$ is real-closed; 2) the extension $k(i)$, where $i^2=-1$, is algebraically closed; or 3) every positive element in $k$ is a square and every polynomial of odd degree over $k$ has a root in $k$. Every formally-real field has a real-closed ordered algebraic extension.

If $k$ is an ordered field, a fundamental sequence can be defined in the usual way (cf. Real number). The set of all fundamental sequences, with proper identification and definition of the operations and transfer of the order, forms an extension $\bar k$ of the field $k$. If $k$ is Archimedean, then $\bar k$ is isomorphic as an ordered field to the real numbers.

References

[1]	N. Bourbaki, "Elements of mathematics" , 2. Algebra. Polynomials and fields. Ordered groups , Hermann (1974) (Translated from French)
[2]	B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German)
[3]	L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963)