Definition:Ordered Field

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

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

If for each $x \in F$, $x > 0 \implies x^{-1} > 0$ then

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

Alternative Definitions
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.