Definition: Ordered Field
We say a field $(F, +, \cdot)$ is ordered, if there exists a strict total order "$<$" on $F$ that fulfills the two properties:
 from \(a > b\) it follows \(a+c > b+c\) (for arbitrary \(c\in F\)),
 from \(a > 0\) and \(b > 0\) it follows \(a\cdot b > 0\).
We call
* $a > 0$ positive,
* $a < 0$ negative,
* $a\ge 0$ nonnegative,
* $a\le 0$ nonpositive.
Examples
 The field of rational numbers $(\mathbb Q,+,\cdot)$
 The field of real numbers $(\mathbb R,+,\cdot)$
Counterexamples
 The field of complex numbers $(\mathbb C,+,\cdot)$
 For a prime number $p,$ the finite field $(\mathbb Z_p,+,\cdot)$
Mentioned in:
Axioms: 1
Definitions: 2 3
Proofs: 4
Propositions: 5 6 7
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References
Adapted from CC BYSA 3.0 Sources:
 Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of Osnabrück