Ordered field

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In mathematics, an ordered field is a field (F,+,*) together with a total order ≤ on F that is compatible with the algebraic operations in the following sense:

• 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.
• Squares are non-negative: 0 ≤ a² for all a in F; in particular 0 < 1.
• Since 0 < 1 + 1 + ... + 1 for any number of summands, the field F has characteristic 0.

It's enough to define whenever a > 0 for every element a, because a > b iff a - b > 0, and all other order relations can be defined based on >.

Every subfield of an ordered field is also an ordered field. The smallest subfield is isomorphic to the rationals (as for any 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.

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 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.

The following conditions are equivalent when R is an ordered field.

• R is a real closed field.
• R is a maximal ordered field.
• R is a maximal real field.

[edit] 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 they contain a square root of -1, which no ordered field can do. 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.