Definition:Ordered Field

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Definition

Let $\left({R, +, \circ, \preceq}\right)$ be an ordered ring.

Let $\left({R, +, \circ}\right)$ be a field.


Then $\left({R, +, \circ, \preceq}\right)$ is an ordered field.


Totally Ordered Field

Let $\left({F, +, \circ, \preceq}\right)$ be an ordered ring.

Let $\left({F, +, \circ}\right)$ be a field.

Let the ordering $\preceq$ be a total ordering.


Then $\left({F, +, \circ, \preceq}\right)$ is a totally ordered field.


Also defined as

The term ordered field is frequently used to refer to what we call a totally ordered field.

Sources defining partially ordered field vary in their definitions. Some require only a field which is an ordered ring, while others impose further restrictions.


Also see

  • Results about ordered fields can be found here.


Sources