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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:

1. from \(a > b\) it follows \(a+c > b+c\) (for arbitrary \(c\in F\)),
2. from \(a > 0\) and \(b > 0\) it follows \(a\cdot b > 0\).

We call * $a > 0$ positive, * $a < 0$ negative, * $a\ge 0$ non-negative, * $a\le 0$ non-positive.

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 BY-SA 3.0 Sources:

1. Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of Osnabrück