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. Historically, the axiomatization of an ordered field was abstracted gradually from the real numbers, by mathematicians including David Hilbert, Otto Hölder and Hans Hahn. In 1926, this grew eventually into the Artin–Schreier theory of ordered fields and formally real fields.

An ordered field necessarily has characteristic 0, i.e., the elements 0, 1, 1 + 1, 1 + 1 + 1, … are all different. This implies that an ordered field necessarily contains an infinite number of elements. Finite fields cannot be ordered.

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. Any Dedekind-complete ordered field is isomorphic to the real numbers. 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. Every ordered field is a formally real field.

Definition

There are two equivalent definitions of an ordered field. Def 1 appeared first historically and is a first-order axiomatization of the ordering ≤ as a binary predicate. Artin and Schreier gave Def 2 in 1926, which axiomatizes the subcollection of nonnegative elements. It subcollection is termed a positive cones (Def 2 below) in 1926. Although Def 2 is higher-order, viewing positive cones as maximal prepositive cones provides a larger context in which field orderings are extremal partial orderings.

Def 1: A total order on F

A field (F,+,*) together with a total order ≤ on F is an ordered field if the order satisfies the following properties:

• if a ≤ b then a + c ≤ b + c
• if 0 ≤ a and 0 ≤ b then 0 ≤ a b

Def 2: A positive cone of F

A prepositive cone of a field F is a subset P ⊂ F that has the following properties:

• For x and y in P, both x+y and xy are in P.
• If x is in F, then x² is in P.
• The element −1 is not in P.

If in addition, the subset F is the union of P and −P, we call P a positive cone of F. The nonzero elements of P are called the positive elements of F.

An ordered field is a field F together with a positive cone P.

Equivalence of the two definitions

Let F be a field. There is a bijection between the field orderings of F and the positive cones of F.

Given a field ordering ≤ as in Def 1, the elements such that x≥0 forms a positive cone of F. Conversely, given a positive cone P of F as in Def 2, one can associate a total ordering ≤_P by setting x≤y to mean y − x ∈ P. This total ordering ≤_P satisfies the properties of Def 1.

Properties of ordered fields

• If x < y and y < z, then x < z. (transitivity)
• If x < y and z > 0, then xz < yz.
• If x < y and x,y > 0, then 1/y < 1/x

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

• Either −a ≤ 0 ≤ a or a ≤ 0 ≤ −a.
• We are allowed to "add inequalities": If a ≤ b and c ≤ d, then a + c ≤ b + d
• We are allowed to "multiply inequalities with positive elements": If a ≤ b and 0 ≤ c, then ac ≤ bc.

• 1 is positive. (Proof: either 1 is positive or −1 is positive. If −1 is positive, then (−1)(−1) = 1 is positive, which is a contradiction)
• An ordered field has characteristic 0. (Since 1 > 0, then 1 + 1 > 0, and 1 + 1 + 1 > 0, etc. If the field had characteristic p > 0, then −1 would be the sum of p − 1 ones, but −1 is not positive). In particular, finite fields cannot be ordered.
• Squares are non-negative. 0 ≤ a² for all a in F. (Follows by a similar argument to 1 > 0)

Every subfield of an ordered field is also an ordered field (inheriting the induced ordering). The smallest subfield is isomorphic to the rationals (as for any other field of characteristic 0), 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. For example, the real numbers form an Archimedean field, but every hyperreal field is non-Archimedean^{[citation needed]}.

An ordered field K is the real number field if it satisfies the axiom of Archimedes and every Cauchy sequence of K converges within K.

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.

Examples of ordered fields

Examples of ordered fields are:

• the rational numbers
• the real algebraic numbers
• the computable numbers
• the real numbers
• the field of real rational functions $\frac {p(x)} {q(x)}\,$, where p(x) and q(x), $q(x) \ne 0\,$ are polynomials with real coefficients, can be made into an ordered field where the polynomial p(x) = x is greater than any constant polynomial, by defining that $\frac {p(x)} {q(x)} > 0\,$ whenever $\frac {p_0} {q_0} > 0\,$, for $p(x) = p_0 x^n + \cdots \mbox{ and } q(x) = q_0 x^m + \cdots\,$. This ordered field is not Archimedean.
• The field of formal Laurent series with real coefficients $\Bbb{R}((x))$, where x is taken to be infinitesimal and positive
• real closed fields
• superreal numbers
• 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.

Which fields can be ordered?

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. (This order is often not uniquely determined.)

Finite fields cannot be turned into ordered fields, because they do not have characteristic 0. The complex numbers also cannot be turned into an ordered field, as −1 is a square (of the imaginary number i) and would thus be positive. Also, the p-adic numbers cannot be ordered, since Q₂ contains a square root of −7 and Q_p (p > 2) contains a square root of 1 − p.

See also

• Ordered ring
• Ordered vector space

References

• Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley Pub. Co., ISBN 978-0-201-55540-0