304 PROPERTIES OF NUMBER SYSTEMS Chapter 9
In the previous article, there was a lack of any reference to ordering of
elements in an arbitrary field. The familiar fact that any two distinct real
numbers a and b satisfy either a < b or a > b is the basis of substantial
portions of the algebra of the real numbers that we learn in high school.
As we will soon see, this part of the theory of the real number field does
not carry over to an arbitrary field, but only to a class of fields known as
orderedfields. The theory of ordered fields will provide a way of distinguish-
ing R, as well as Q, from C.
DEFINITION 1
An ordered field (F, 8) consists of a field (F, +, .), together with a nonempty
subset 9 of F satisfying:
(a) For all x, YE F, if x, YE 9, then x + ye 9
(b) Forallx,y~F, ifx,y€P,thenxy€9
(c) For any x E F, one and only one of the following three statements is true:
(i) XEB
(ii) - x E 9
(iii) x = 0
The subset 9 is called the positive part of the ordered field (F, S), and an ele-
ment x E 8 is called a positive element of F. If - x E 8, x is said to be a negative
element of (F, 9).Conditions (a) and (b) state that 9 is closed under the addition and
multiplication operations of the field F. Condition (c) is called trichotomy.
When there is no danger of ambiguity, we sometimes refer simply to the
ordered field F, rather than the ordered field (F, 9).
Our first theorem about ordered fields is not only important in its own
right, but contains all that is needed to show that the complex number field
does not admit an ordering, that is, cannot possibly be ordered. Remember,
R is, by definition, an ordered field.THEOREM 1
Let (F, 9) be an ordered field. Then:
(a) 1 E 9
(b) If XE~, then x-'E~
(c) If x E F and x # 0, then x2 E 9
Proof (a) Since 1 # 0, then either 1 E 9 or - 1 E 9, by (c) of Definition 1.
Suppose - 1 E 9 and let a E 9. Then (- l)a E 9 by (b) of Definition 1.
But (- l)a = -a by Theorem 3, Article 9.1, so that -a E 9. But a E 9
and -a E 9 contradicts (c) of Definition 1.
(b) Assume x~9 SO that x#O and x-' #O. If x-'#9, then- x- ' E 9 by (c) of Definition 1. But then, (x)(- x- ') = - (XX- ') =
- 1 E 9, a contradiction since 1 E 9, by (a).