## Ordered Field

An **ordered field** is a field *F* together with a positive cone *P*.

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### Some articles on ordered field:

... A real closed

**field**is a

**field**F in which any of the following equivalent conditions are true F is elementarily equivalent to the real numbers ... it has the same first-order properties as the reals any sentence in the first-order language of

**fields**is true in F if and only if it is true in the reals ... There is a total order on F making it an

**ordered field**such that, in this ordering, every positive element of F is a square in F and any polynomial of odd degree with ...

**Ordered Field**- Harrison Topology

... The Harrison topology is a topology on the set of orderings XF of a formally real field F ... Each order can be regarded as a multiplicative group homomorphism from F* onto ±1 ...

... See, for example, algebraically closed

**field**or compactification ... be described equivalently as either the completeness of R as metric space or as a partially

**ordered**set (see below) ... In order theory and related

**fields**such as lattice and domain theory, completeness generally refers to the existence of certain suprema or infima of ...

**Ordered Field**

... Every linearly

**ordered field**K contains (an isomorphic copy of) the rationals as an

**ordered**subfield, namely the subfield generated by the multiplicative unit 1 of K, which in turn ... The following are equivalent characterizations of Archimedean

**fields**in terms of these substructures ... not the case when there exist infinite elements.) Thus an Archimedean

**field**is one whose natural numbers grow without bound ...

**Ordered Field**

... For an example of an

**ordered field**that is not Archimedean, take the

**field**of rational functions with real coefficients ... coefficient of the denominator is positive.) To make this an

**ordered field**, one must assign an ordering compatible with the addition and multiplication ... Therefore, 1/x is an infinitesimal in this

**field**...

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