# 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 $\struct {F, +, \circ, \preceq}$ be an ordered ring.

Let $\struct {F, +, \circ}$ be a field.

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

Then $\struct {F, +, \circ, \preceq}$ 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

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 23$ - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers