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# 2.6 Ordered Fields

2.100   Definition (Ordered field axioms.) An ordered field is a pair where is a field, and is a subset of satisfying the conditions
1. For all , .
2. For all , .
3. (Trichotomy) For all , exactly one of the statements

is true. The set is called the set of positive elements of . A field is orderable if it has a subset such that 1), 2) and 3) are satisfied.

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, .

2.105   Example. The field is not orderable.

First Proof: If there were a subset of such that were an ordered field, we would have . But in , so and , which contradicts trichotomy.

Second Proof: If were an ordered field, we would have , so , so , so so . This contradicts trichotomy.

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
1. and
2. and
3. and
These laws together with the axiom

are called the laws of signs.

2.117   Notation. Let be an ordered field, and let be non-zero 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 .

2.127   Theorem. Let be an ordered field and let be elements of with and . Then
 (2.128)

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