Ordered field
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 cannot be written as a sum of squares. A field for which
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 of an ordered field
is said to be ordered if
is an ordered field containing
as ordered subfield. This takes place precisely when
cannot be written as a sum of elements of the form
, where
,
and
. 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
are equivalent: 1)
is real-closed; 2) the extension
, where
, is algebraically closed; or 3) every positive element in
is a square and every polynomial of odd degree over
has a root in
. Every formally-real field has a real-closed ordered algebraic extension.
If 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
of the field
. If
is Archimedean, then
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) |
Ordered field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ordered_field&oldid=16502