Definition:Ordered Field

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.