Ordered field
Basic examples of ordered fields are the rational numbers and the real numbers, both with their standard orderings.
Every Dedekind-complete ordered field is isomorphic to the reals.
Squares are necessarily non-negative in an ordered field.
This implies that the complex numbers cannot be ordered since the square of the imaginary unit i is −1 (which is negative in any ordered field).
Historically, the axiomatization of an ordered field was abstracted gradually from the real numbers, by mathematicians including David Hilbert, Otto Hölder and Hans Hahn.
There are two equivalent common definitions of an ordered field.
Artin and Schreier gave the definition in terms of positive cone in 1926, which axiomatizes the subcollection of nonnegative elements.
Although the latter is higher-order, viewing positive cones as maximal prepositive cones provides a larger context in which field orderings are extremal partial orderings.
A prepositive cone or preordering of a field
form a subgroup of the multiplicative group of
are precisely the intersections of families of positive cones on
The positive cones are the maximal preorderings.
Given a field ordering ≤ as in the first definition, the set of elements such that
as in the second definition, one can associate a total ordering
Examples of ordered fields are: The surreal numbers form a proper class rather than a set, but otherwise obey the axioms of an ordered field.
Every ordered field can be embedded into the surreal numbers.
If every element of an ordered field lies between two elements of its rational subfield, then the field is said to be Archimedean.
[4] An ordered field F is isomorphic to the real number field R if and only if every non-empty subset of F with an upper bound in F has a least upper bound in F. This property implies that the field is Archimedean.
Vector spaces (particularly, n-spaces) over an ordered field exhibit some special properties and have some specific structures, namely: orientation, convexity, and positively-definite inner product.
See Real coordinate space#Geometric properties and uses for discussion of those properties of Rn, which can be generalized to vector spaces over other ordered fields.
Every ordered field is a formally real field, i.e., 0 cannot be written as a sum of nonzero squares.
[2][3] Conversely, every formally real field can be equipped with a compatible total order, that will turn it into an ordered field.
The complex numbers also cannot be turned into an ordered field, as −1 is a square of the imaginary unit i.
Also, the p-adic numbers cannot be ordered, since according to Hensel's lemma Q2 contains a square root of −7, thus 12 + 12 + 12 + 22 + √−72 = 0, and Qp (p > 2) contains a square root of 1 − p, thus (p − 1)⋅12 + √1 − p2 = 0.
[6] If F is equipped with the order topology arising from the total order ≤, then the axioms guarantee that the operations + and × are continuous, so that F is a topological field.
The Harrison topology is a topology on the set of orderings XF of a formally real field F. Each order can be regarded as a multiplicative group homomorphism from F∗ onto ±1.
form a subbasis for the Harrison topology.
The product is a Boolean space (compact, Hausdorff and totally disconnected), and XF is a closed subset, hence again Boolean.
[7][8] A fan on F is a preordering T with the property that if S is a subgroup of index 2 in F∗ containing T − {0} and not containing −1 then S is an ordering (that is, S is closed under addition).
[9] A superordered field is a totally real field in which the set of sums of squares forms a fan.
