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. An example of an ordered field is the field of real numbers. This concept was introduced by Emil Artin in 1927.
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)
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