Ordered field

In mathematics, an ordered field is a field together with a total ordering of its elements that agrees in a certain sense with the field operations. This concept was introduced by Emil Artin in 1927.

Definition

There are two equivalent definitions, depending on which properties one takes as the definition for an ordered field.

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

It follows from these axioms that 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.

Def 2: An ordering on F

An ordering of a field F is a subset P ⊂ F that has the following properties:

• F is the disjoint union of P, −P, and the element 0. That is, for each x ∈ F, exactly one of the following conditions is true: x = 0, x ∈ P or −x ∈ P.
• For x and y in P, both x+y and xy are in P.

The elements of the subset P are called the positive elements of F.

We next define x < y to mean that y − x ∈ P (so that y − x > 0 in a sense). This relation satisfies the expected properties:

• 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

The statement x ≤ y will mean that either x < y or x = y.

Properties of ordered fields

• 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. 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 the 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 , where p(x) and q(x), 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 whenever , for . This ordered field is not Archimedean.
• The field of formal Laurent series with real coefficients , 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.

References

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