Definition:Totally Ordered Field

Definition

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 known as

This is often referred to as an ordered field.

Also see

• Field is Totally Ordered iff Ordered Compatibly with Ring Structure
• Properties of Ordered Field

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $23$. The Field of Rational Numbers