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2.101
Example.
The rational numbers
form an ordered field, where
denotes
the familiar set of positive rationals.
2.102
Notation (.)
Let
be an ordered field. We let
We call
the set of
negative elements in
. Thus
and
We can restate the Trichotomy axiom as: For all
, exactly one of the
statements
is true.
2.103
Theorem.
Let be an ordered field. Then for all
, .
Proof: Since , we know or . Now
and
2.104
Corollary.
In any ordered field, .
2.106
Remark.
The method used in the second proof above shows that none of the fields
are orderable.
2.107
Definition ()
Let
be an ordered field, and let
. We define
2.108
Remark.
In any ordered field
:
2.109
Exercise.
Let
be an ordered field, and let
. Show that exactly one of
the statements
is true.
2.110
Theorem (Transitivity of .)
Let be an ordered field. Then for all ,
Proof: For all we have
2.111
Exercise (Addition of inequalities.)
Let
be an ordered field, and let
. Show that
and
2.112
Exercise.
Let
be an ordered field, and let
. Show that
2.113
Notation.
Let
be an ordered field, and let
. We use notation like

(2.114) 
to mean
and
and
, and similarly we write

(2.115) 
to mean
and
and
. By transitivity of
and of
, you can
conclude
from (
2.114), and you can conclude
and
from
(
2.115). A chain of inequalities involving both
and
shows bad
style, so you should not write
2.116
Exercise (Laws of signs.)
Let
be an ordered field, and let
. Show that
 and
 and
 and
These laws together with the axiom
are called the
laws of signs.
2.117
Notation.
Let
be an ordered field, and let
be nonzero elements of
. We say
and
have the same sign if either (
are both in
) or (
are both in
). Otherwise we say
and have opposite signs.
2.118
Corollary (of the law of signs.)Let be an ordered field and let
. Then
2.119
Notation.
I will now start to use the convention that `` let
be an ordered field" means ``let
be an ordered field''; i.e.,
the set of positive elements of
is assumed to be called
.
2.120
Exercise.
Let
be an ordered field and let
. Prove that
2.121
Theorem (Multiplication of inequalities.)Let be an ordered field and let be elements of . Then
Proof: By the previous exercise we have
Hence, by transitivity of ,
2.122
Exercise.
Let
be an ordered field, and let
. Show that
and
have the same sign.
2.123
Exercise.
A
Let
be an ordered field, and let
.
Under what conditions (if any) can you say that
Under what conditions (if any) can you say that
2.124
Definition (Square root.)
Let
be a field, and let
. A square root for
is any element
of
such that
.
2.125
Examples.
In
, the square roots of
are
and
.
In an ordered field , no element in has a square root.
In
, there is no square root of . (See theorem 3.45
for a proof.)
2.126
Theorem.
Let be an ordered field and let . If has a square root, then it
has exactly two square roots, one in and one in , so if has a square
root, it has a unique positive square root.
Proof: Suppose has a square root . Then , since . If
is any square root of , then , so, as we saw in theorem 2.95,
or . By trichotomy, one of is in , and the other is in
.
Proof: Let be elements of . Then , unless ,
so
. Hence
2.129
Remark.
The implication (
2.128) is also true when
is replaced by
in both
positions. I'll leave this to you to check.
Next: 2.7 Absolute Value
Up: 2. Fields
Previous: 2.5 Subtraction and Division
Index