Definition:Ordered Field
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Definition
Let $\struct {R, +, \circ, \preceq}$ be an ordered ring.
Let $\struct {R, +, \circ}$ be a field.
Then $\struct {R, +, \circ, \preceq}$ 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): Chapter $\text {IV}$: Rings and Fields: $23$. The Field of Rational Numbers
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): ordered field