# ordered field meaning

ENWOrdered field

- In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations.
- An ordered field necessarily has characteristic 0, all natural numbers, i.e. the elements 0, 1, 1 + 1, 1 + 1 + 1, … are distinct. This implies that an ordered field necessarily contains an infinite number of elements: a finite field cannot be ordered.
- Every subfield of an ordered field is also an ordered field in the inherited order. Every ordered field contains an ordered subfield that is isomorphic to the rational numbers. Any Dedekind-complete ordered field is isomorphic to the real numbers.

- NounPLordered fields
- (algebra) A field which has an order relation satisfying these properties: trichotomy, transitivity, preservation of an inequality when the same element is added to both sides, and preservation of an inequality when the same strictly positive element is multiplied to both sides.

- (algebra) A field which has an order relation satisfying these properties: trichotomy, transitivity, preservation of an inequality when the same element is added to both sides, and preservation of an inequality when the same strictly positive element is multiplied to both sides.

## Definition of __ordered field__ in English Dictionary

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Source: Wiktionary

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