---
tag: 010JCV9J4E
kind: definition
name: Ordered field
msc: [12J15]
defines: 0P3VCBDEE2
sources:
  - url: https://en.wikipedia.org/wiki/Ordered_field
    locator: Total order
---

## Definition

An ordered field is a field $F$ with a total order $\le$ on $F$.
The order must have these two properties for all $a$, $b$ and $c$ in $F$:

1. if $a \le b$, then $a + c \le b + c$;
2. if $0 \le a$ and $0 \le b$, then $0 \le a b$.

## Remarks

In an ordered field, each square is nonnegative: $a^2 \ge 0$ for each $a$. In particular $1 = 1^2 > 0$ and $-1 < 0$.
An equivalent definition uses a positive cone $P$. This is a subset of $F$ that is closed under addition and multiplication.
Also, $F$ must be the disjoint union of $P$, $\{0\}$ and $-P$.
The field $\mathbb{C}$ of complex numbers [[has no such order|01CJRJKDAA]].

## Explanations

### L1 human

An ordered field is a number system where you can add, multiply, divide and compare numbers.
The rules of comparison are the usual ones.
If you add the same number to both sides of an inequality, the inequality stays true.
The product of two positive numbers is positive.