Ordered field

In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Basic examples of ordered fields are the rational numbers and the real numbers, both with their standard orderings.
Every subfield of an ordered field is also an ordered field in the inherited order. Every ordered field contains an ordered subfield that is isomorphic to the rational numbers. 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 . Finite fields cannot be ordered.
Historically, the axiomatization of an ordered field was abstracted gradually from the real numbers, by mathematicians including David Hilbert, Otto Hölder and Hans Hahn. This grew eventually into the Artin–Schreier theory of ordered fields and formally real fields.

Definitions

There are two equivalent common definitions of an ordered field. The definition of total order appeared first historically and is a first-order axiomatization of the ordering as a binary predicate. 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 prepositive cones provides a larger context in which field orderings are partial orderings.

Total order

A field together with a total order on is an if the order satisfies the following properties for all

if then and
if and then

As usual, we write for and. The notations and stand for and, respectively. Elements with are called positive.

Positive cone

A ' or preordering of a field is a subset that has the following properties:
For and in both and are in
If then In particular, and
The element is not in
A ' is a field equipped with a preordering Its non-zero elements form a subgroup of the multiplicative group of
If in addition, the set is the union of and we call a positive cone of The non-zero elements of are called the positive elements of
An ordered field is a field together with a positive cone
The preorderings on are precisely the intersections of families of positive cones on The positive cones are the maximal preorderings.

Equivalence of the two definitions

Let be a field. There is a bijection between the field orderings of and the positive cones of
Given a field ordering ≤ as in the first definition, the set of elements such that forms a positive cone of Conversely, given a positive cone of as in the second definition, one can associate a total ordering on by setting to mean This total ordering satisfies the properties of the first definition.

Examples of ordered fields

Examples of ordered fields are:

the field of rational numbers with its standard ordering ;
the field of real numbers with its standard ordering ;
any subfield of an ordered field, such as the real algebraic numbers or the computable numbers, becomes an ordered field by restricting the ordering to the subfield;
the field of rational functions, where and are polynomials with rational coefficients and, can be made into an ordered field by fixing a real transcendental number and defining if and only if. This is equivalent to embedding into via and restricting the ordering of to an ordering of the image of. In this fashion, we get many different orderings of.
the field of rational functions, where and are polynomials with real coefficients and, can be made into an ordered field by defining to mean that, where and are the leading coefficients of and, respectively. Equivalently: for rational functions we have if and only if for all sufficiently large. In this ordered field the polynomial is greater than any constant polynomial and the ordered field is not Archimedean.
The field of formal Laurent series with real coefficients, where x is taken to be infinitesimal and positive
the transseries
real closed fields
the superreal numbers
the hyperreal numbers

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.

Properties of ordered fields

For every a, b, c, d in F:

Either −a ≤ 0 ≤ a or a ≤ 0 ≤ −a.
One can "add inequalities": if a ≤ b and c ≤ d, then a + c ≤ b + d.
One can "multiply inequalities with positive elements": if a ≤ b and 0 ≤ c, then ac ≤ bc.
"Multiplying with negatives flips an inequality": if a ≤ b and c ≤ 0, then ac ≥ bc.
If a < b and a, b > 0, then 1/b < 1/a.
Squares are non-negative: 0 ≤ a² for all a in F. In particular, since 1=1², it follows that 0 ≤ 1. Since 0 ≠ 1, we conclude 0 < 1.
An ordered field has characteristic 0. In particular, finite fields cannot be ordered.
Every non-trivial sum of squares is nonzero. Equivalently:

Every subfield of an ordered field is also an ordered field. The smallest subfield is isomorphic to the rationals, and the order on this rational subfield is the same as the order of the rationals themselves.
If every element of an ordered field lies between two elements of its rational subfield, then the field is said to be Archimedean. Otherwise, such field is a non-Archimedean ordered field and contains infinitesimals as well as infinite elements larger than any rational. For example, the real numbers form an Archimedean field, but hyperreal numbers form a non-Archimedean field, because it extends real numbers with elements greater than any standard natural number.
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 over an ordered field

s 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.

Orderability of fields

Every ordered field is a formally real field, i.e., 0 cannot be written as a sum of nonzero squares.
Conversely, every formally real field can be equipped with a compatible total order, that will turn it into an ordered field. The proof uses Zorn's lemma.
Finite fields and more generally fields of positive characteristic cannot be turned into ordered fields, as shown above. 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 + ² = 0, and Q_p contains a square root of 1 − p, thus ⋅1² + ² = 0.

Topology induced by the order

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.

Harrison topology

The Harrison topology is a topology on the set of orderings X_F of a formally real field F. Each order can be regarded as a multiplicative group homomorphism from F^∗ onto ±1. Giving ±1 the discrete topology and ±1^F the product topology induces the subspace topology on X_F. The Harrison sets form a subbasis for the Harrison topology. The product is a Boolean space, and X_F is a closed subset, hence again Boolean.

Fans and superordered fields

A fan on F is a preordering T with the property that if S is a subgroup of index 2 in F^∗ containing T − and not containing −1 then S is an ordering. A superordered field is a totally real field in which the set of sums of squares forms a fan.