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.
Articles by others on the same topic
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 \( b \leq a \).