Definition:Ordered Field

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

If $$\left({R, +, \circ}\right)$$ is a field, then $$\left({R, +, \circ; \le}\right)$$ is an ordered field.

Totally Ordered Field
If $$\le$$ is a total ordering, then $$\left({R, +, \circ; \le}\right)$$ is a totally ordered field.