An ordered field is a field \( F \) equipped with a total order \( \leq \) that is compatible with the field operations. This means that the order satisfies the following properties: 1. **Totality**: For any two elements \( a, b \in F \), one of the following holds: \( a \leq b \) or \( b \leq a \).
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An ordered field is a field with a total order compatible with its arithmetic: implies , and imply . The rational numbers and real numbers have their usual ordered-field structures.
Every nonzero square is positive. Indeed, either or , and multiplying the positive element by itself gives . In particular , so every sum is nonzero and an ordered field has characteristic zero. The complex numbers cannot admit such an order because would force .
An ordered field need not satisfy order completeness or the Archimedean property. In the rational-function field , declaring a nonzero rational function positive when it is positive for all sufficiently large real defines an order in which exceeds every real constant. Order compatibility and completeness are separate assumptions.