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logo1 Positive Rational Numbers Ordered Fields

Ordered Fields

Bernd Schr¨

• der

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

Theorem.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Theorem. The subset P :=
• [(n,d)] : n ∈ N,d ∈ N
• ⊆ Q of the

rational numbers is called the set of positive rational numbers.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Theorem. The subset P :=
• [(n,d)] : n ∈ N,d ∈ N
• ⊆ Q of the

rational numbers is called the set of positive rational

• numbers. It has the following properties.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Theorem. The subset P :=
• [(n,d)] : n ∈ N,d ∈ N
• ⊆ Q of the

rational numbers is called the set of positive rational

• numbers. It has the following properties.
• 1. For all x,y ∈ P, we have x+y ∈ P and xy ∈ P,

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Theorem. The subset P :=
• [(n,d)] : n ∈ N,d ∈ N
• ⊆ Q of the

rational numbers is called the set of positive rational

• numbers. It has the following properties.
• 1. For all x,y ∈ P, we have x+y ∈ P and xy ∈ P,
• 2. For all x ∈ Q, exactly one of the following three properties

holds.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Theorem. The subset P :=
• [(n,d)] : n ∈ N,d ∈ N
• ⊆ Q of the

rational numbers is called the set of positive rational

• numbers. It has the following properties.
• 1. For all x,y ∈ P, we have x+y ∈ P and xy ∈ P,
• 2. For all x ∈ Q, exactly one of the following three properties

holds. Either x ∈ P

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Theorem. The subset P :=
• [(n,d)] : n ∈ N,d ∈ N
• ⊆ Q of the

rational numbers is called the set of positive rational

• numbers. It has the following properties.
• 1. For all x,y ∈ P, we have x+y ∈ P and xy ∈ P,
• 2. For all x ∈ Q, exactly one of the following three properties

holds. Either x ∈ P or −x ∈ P

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Theorem. The subset P :=
• [(n,d)] : n ∈ N,d ∈ N
• ⊆ Q of the

rational numbers is called the set of positive rational

• numbers. It has the following properties.
• 1. For all x,y ∈ P, we have x+y ∈ P and xy ∈ P,
• 2. For all x ∈ Q, exactly one of the following three properties

holds. Either x ∈ P or −x ∈ P or x = 0.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Theorem. The subset P :=
• [(n,d)] : n ∈ N,d ∈ N
• ⊆ Q of the

rational numbers is called the set of positive rational

• numbers. It has the following properties.
• 1. For all x,y ∈ P, we have x+y ∈ P and xy ∈ P,
• 2. For all x ∈ Q, exactly one of the following three properties

holds. Either x ∈ P or −x ∈ P or x = 0. Proof.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Theorem. The subset P :=
• [(n,d)] : n ∈ N,d ∈ N
• ⊆ Q of the

rational numbers is called the set of positive rational

• numbers. It has the following properties.
• 1. For all x,y ∈ P, we have x+y ∈ P and xy ∈ P,
• 2. For all x ∈ Q, exactly one of the following three properties

holds. Either x ∈ P or −x ∈ P or x = 0.

• Proof. Good exercise.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Theorem. The subset P :=
• [(n,d)] : n ∈ N,d ∈ N
• ⊆ Q of the

rational numbers is called the set of positive rational

• numbers. It has the following properties.
• 1. For all x,y ∈ P, we have x+y ∈ P and xy ∈ P,
• 2. For all x ∈ Q, exactly one of the following three properties

holds. Either x ∈ P or −x ∈ P or x = 0.

• Proof. Good exercise.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

Definition.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Definition. A field (F,+,·) is called a totally ordered field iff

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Definition. A field (F,+,·) is called a totally ordered field iff

there is a subset F+ ⊆ F

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Definition. A field (F,+,·) is called a totally ordered field iff

there is a subset F+ ⊆ F so that

• 1. For all x,y ∈ F+ we have x+y ∈ F+ and xy ∈ F+, and

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Definition. A field (F,+,·) is called a totally ordered field iff

there is a subset F+ ⊆ F so that

• 1. For all x,y ∈ F+ we have x+y ∈ F+ and xy ∈ F+, and
• 2. 0 ∈ F+, and

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Definition. A field (F,+,·) is called a totally ordered field iff

there is a subset F+ ⊆ F so that

• 1. For all x,y ∈ F+ we have x+y ∈ F+ and xy ∈ F+, and
• 2. 0 ∈ F+, and
• 3. For all x ∈ F\{0}, either x ∈ F+ or −x ∈ F+ holds.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Definition. A field (F,+,·) is called a totally ordered field iff

there is a subset F+ ⊆ F so that

• 1. For all x,y ∈ F+ we have x+y ∈ F+ and xy ∈ F+, and
• 2. 0 ∈ F+, and
• 3. For all x ∈ F\{0}, either x ∈ F+ or −x ∈ F+ holds.

The subset F+ is also called the positive cone of the totally

• rdered field.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Definition. A field (F,+,·) is called a totally ordered field iff

there is a subset F+ ⊆ F so that

• 1. For all x,y ∈ F+ we have x+y ∈ F+ and xy ∈ F+, and
• 2. 0 ∈ F+, and
• 3. For all x ∈ F\{0}, either x ∈ F+ or −x ∈ F+ holds.

The subset F+ is also called the positive cone of the totally

• rdered field. The elements of F+ are called nonnegative

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Definition. A field (F,+,·) is called a totally ordered field iff

there is a subset F+ ⊆ F so that

• 1. For all x,y ∈ F+ we have x+y ∈ F+ and xy ∈ F+, and
• 2. 0 ∈ F+, and
• 3. For all x ∈ F\{0}, either x ∈ F+ or −x ∈ F+ holds.

The subset F+ is also called the positive cone of the totally

• rdered field. The elements of F+ are called nonnegative, the

elements of F+ \{0} are called positive

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Definition. A field (F,+,·) is called a totally ordered field iff

there is a subset F+ ⊆ F so that

• 1. For all x,y ∈ F+ we have x+y ∈ F+ and xy ∈ F+, and
• 2. 0 ∈ F+, and
• 3. For all x ∈ F\{0}, either x ∈ F+ or −x ∈ F+ holds.

The subset F+ is also called the positive cone of the totally

• rdered field. The elements of F+ are called nonnegative, the

elements of F+ \{0} are called positive, and the elements x ∈ F\{0} so that −x ∈ F+ are called negative.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

What do we already know about totally ordered fields in general and about Q in particular?

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

What do we already know about totally ordered fields in general and about Q in particular?

• 1. There is an order relation x ≤ y iff y−x ∈ F+.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

What do we already know about totally ordered fields in general and about Q in particular?

• 1. There is an order relation x ≤ y iff y−x ∈ F+.
• 2. The order relation works as we expect it to work

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

What do we already know about totally ordered fields in general and about Q in particular?

• 1. There is an order relation x ≤ y iff y−x ∈ F+.
• 2. The order relation works as we expect it to work

(multiplication with positive numbers preserves the inequality

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

What do we already know about totally ordered fields in general and about Q in particular?

• 1. There is an order relation x ≤ y iff y−x ∈ F+.
• 2. The order relation works as we expect it to work

(multiplication with positive numbers preserves the inequality, multiplication with negative numbers reverses it).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

What do we already know about totally ordered fields in general and about Q in particular?

• 1. There is an order relation x ≤ y iff y−x ∈ F+.
• 2. The order relation works as we expect it to work

(multiplication with positive numbers preserves the inequality, multiplication with negative numbers reverses it).

• 3. There is an absolute value function that has the properties

we expect it to have

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

What do we already know about totally ordered fields in general and about Q in particular?

• 1. There is an order relation x ≤ y iff y−x ∈ F+.
• 2. The order relation works as we expect it to work

(multiplication with positive numbers preserves the inequality, multiplication with negative numbers reverses it).

• 3. There is an absolute value function that has the properties

we expect it to have (it’s nonnegative

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

What do we already know about totally ordered fields in general and about Q in particular?

• 1. There is an order relation x ≤ y iff y−x ∈ F+.
• 2. The order relation works as we expect it to work

(multiplication with positive numbers preserves the inequality, multiplication with negative numbers reverses it).

• 3. There is an absolute value function that has the properties

we expect it to have (it’s nonnegative, it’s zero iff the element is zero

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

What do we already know about totally ordered fields in general and about Q in particular?

• 1. There is an order relation x ≤ y iff y−x ∈ F+.
• 2. The order relation works as we expect it to work

(multiplication with positive numbers preserves the inequality, multiplication with negative numbers reverses it).

• 3. There is an absolute value function that has the properties

we expect it to have (it’s nonnegative, it’s zero iff the element is zero, it factors through products

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

What do we already know about totally ordered fields in general and about Q in particular?

• 1. There is an order relation x ≤ y iff y−x ∈ F+.
• 2. The order relation works as we expect it to work

(multiplication with positive numbers preserves the inequality, multiplication with negative numbers reverses it).

• 3. There is an absolute value function that has the properties

we expect it to have (it’s nonnegative, it’s zero iff the element is zero, it factors through products and it obeys the triangular inequality).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

Theorem.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Theorem. Let (F,+,·) be a totally ordered field and let x,y ∈ F

be so that 0 < x ≤ y.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Theorem. Let (F,+,·) be a totally ordered field and let x,y ∈ F

be so that 0 < x ≤ y. Then 0 < y−1 ≤ x−1.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Theorem. Let (F,+,·) be a totally ordered field and let x,y ∈ F

be so that 0 < x ≤ y. Then 0 < y−1 ≤ x−1. Proof.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Theorem. Let (F,+,·) be a totally ordered field and let x,y ∈ F

be so that 0 < x ≤ y. Then 0 < y−1 ≤ x−1.

• Proof. We first prove that x > 0 implies x−1 > 0.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Theorem. Let (F,+,·) be a totally ordered field and let x,y ∈ F

be so that 0 < x ≤ y. Then 0 < y−1 ≤ x−1.

• Proof. We first prove that x > 0 implies x−1 > 0. Suppose for a

contradiction that x−1 ≤ 0.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Theorem. Let (F,+,·) be a totally ordered field and let x,y ∈ F

be so that 0 < x ≤ y. Then 0 < y−1 ≤ x−1.

• Proof. We first prove that x > 0 implies x−1 > 0. Suppose for a

contradiction that x−1 ≤ 0. Then x−1 < 0.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Theorem. Let (F,+,·) be a totally ordered field and let x,y ∈ F

be so that 0 < x ≤ y. Then 0 < y−1 ≤ x−1.

• Proof. We first prove that x > 0 implies x−1 > 0. Suppose for a

contradiction that x−1 ≤ 0. Then x−1 < 0. Now x > 0 implies that 1 =

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Theorem. Let (F,+,·) be a totally ordered field and let x,y ∈ F

be so that 0 < x ≤ y. Then 0 < y−1 ≤ x−1.

• Proof. We first prove that x > 0 implies x−1 > 0. Suppose for a

contradiction that x−1 ≤ 0. Then x−1 < 0. Now x > 0 implies that 1 = x·x−1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Theorem. Let (F,+,·) be a totally ordered field and let x,y ∈ F

be so that 0 < x ≤ y. Then 0 < y−1 ≤ x−1.

• Proof. We first prove that x > 0 implies x−1 > 0. Suppose for a

contradiction that x−1 ≤ 0. Then x−1 < 0. Now x > 0 implies that 1 = x·x−1 ≤ 0·x−1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Theorem. Let (F,+,·) be a totally ordered field and let x,y ∈ F

be so that 0 < x ≤ y. Then 0 < y−1 ≤ x−1.

• Proof. We first prove that x > 0 implies x−1 > 0. Suppose for a

contradiction that x−1 ≤ 0. Then x−1 < 0. Now x > 0 implies that 1 = x·x−1 ≤ 0·x−1 = 0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Theorem. Let (F,+,·) be a totally ordered field and let x,y ∈ F

be so that 0 < x ≤ y. Then 0 < y−1 ≤ x−1.

• Proof. We first prove that x > 0 implies x−1 > 0. Suppose for a

contradiction that x−1 ≤ 0. Then x−1 < 0. Now x > 0 implies that 1 = x·x−1 ≤ 0·x−1 = 0, contradiction.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Theorem. Let (F,+,·) be a totally ordered field and let x,y ∈ F

be so that 0 < x ≤ y. Then 0 < y−1 ≤ x−1.

• Proof. We first prove that x > 0 implies x−1 > 0. Suppose for a

contradiction that x−1 ≤ 0. Then x−1 < 0. Now x > 0 implies that 1 = x·x−1 ≤ 0·x−1 = 0, contradiction. Now let 0 < x ≤ y.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Theorem. Let (F,+,·) be a totally ordered field and let x,y ∈ F

be so that 0 < x ≤ y. Then 0 < y−1 ≤ x−1.

• Proof. We first prove that x > 0 implies x−1 > 0. Suppose for a

contradiction that x−1 ≤ 0. Then x−1 < 0. Now x > 0 implies that 1 = x·x−1 ≤ 0·x−1 = 0, contradiction. Now let 0 < x ≤ y. We know that x−1,y−1 > 0.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Theorem. Let (F,+,·) be a totally ordered field and let x,y ∈ F

be so that 0 < x ≤ y. Then 0 < y−1 ≤ x−1.

• Proof. We first prove that x > 0 implies x−1 > 0. Suppose for a

contradiction that x−1 ≤ 0. Then x−1 < 0. Now x > 0 implies that 1 = x·x−1 ≤ 0·x−1 = 0, contradiction. Now let 0 < x ≤ y. We know that x−1,y−1 > 0. Therefore x−1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Theorem. Let (F,+,·) be a totally ordered field and let x,y ∈ F

be so that 0 < x ≤ y. Then 0 < y−1 ≤ x−1.

• Proof. We first prove that x > 0 implies x−1 > 0. Suppose for a

contradiction that x−1 ≤ 0. Then x−1 < 0. Now x > 0 implies that 1 = x·x−1 ≤ 0·x−1 = 0, contradiction. Now let 0 < x ≤ y. We know that x−1,y−1 > 0. Therefore x−1 = x−1 ·1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Theorem. Let (F,+,·) be a totally ordered field and let x,y ∈ F

be so that 0 < x ≤ y. Then 0 < y−1 ≤ x−1.

• Proof. We first prove that x > 0 implies x−1 > 0. Suppose for a

contradiction that x−1 ≤ 0. Then x−1 < 0. Now x > 0 implies that 1 = x·x−1 ≤ 0·x−1 = 0, contradiction. Now let 0 < x ≤ y. We know that x−1,y−1 > 0. Therefore x−1 = x−1 ·1 = x−1yy−1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Theorem. Let (F,+,·) be a totally ordered field and let x,y ∈ F

be so that 0 < x ≤ y. Then 0 < y−1 ≤ x−1.

• Proof. We first prove that x > 0 implies x−1 > 0. Suppose for a

contradiction that x−1 ≤ 0. Then x−1 < 0. Now x > 0 implies that 1 = x·x−1 ≤ 0·x−1 = 0, contradiction. Now let 0 < x ≤ y. We know that x−1,y−1 > 0. Therefore x−1 = x−1 ·1 = x−1yy−1 ≥ x−1xy−1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Theorem. Let (F,+,·) be a totally ordered field and let x,y ∈ F

be so that 0 < x ≤ y. Then 0 < y−1 ≤ x−1.

• Proof. We first prove that x > 0 implies x−1 > 0. Suppose for a

contradiction that x−1 ≤ 0. Then x−1 < 0. Now x > 0 implies that 1 = x·x−1 ≤ 0·x−1 = 0, contradiction. Now let 0 < x ≤ y. We know that x−1,y−1 > 0. Therefore x−1 = x−1 ·1 = x−1yy−1 ≥ x−1xy−1 = y−1.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Theorem. Let (F,+,·) be a totally ordered field and let x,y ∈ F

be so that 0 < x ≤ y. Then 0 < y−1 ≤ x−1.

• Proof. We first prove that x > 0 implies x−1 > 0. Suppose for a

contradiction that x−1 ≤ 0. Then x−1 < 0. Now x > 0 implies that 1 = x·x−1 ≤ 0·x−1 = 0, contradiction. Now let 0 < x ≤ y. We know that x−1,y−1 > 0. Therefore x−1 = x−1 ·1 = x−1yy−1 ≥ x−1xy−1 = y−1.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Theorem. Let (F,+,·) be a totally ordered field and let x,y ∈ F

be so that 0 < x ≤ y. Then 0 < y−1 ≤ x−1.

• Proof. We first prove that x > 0 implies x−1 > 0. Suppose for a

contradiction that x−1 ≤ 0. Then x−1 < 0. Now x > 0 implies that 1 = x·x−1 ≤ 0·x−1 = 0, contradiction. Now let 0 < x ≤ y. We know that x−1,y−1 > 0. Therefore x−1 = x−1 ·1 = x−1yy−1 ≥ x−1xy−1 = y−1. Theorem.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Theorem. Let (F,+,·) be a totally ordered field and let x,y ∈ F

be so that 0 < x ≤ y. Then 0 < y−1 ≤ x−1.

• Proof. We first prove that x > 0 implies x−1 > 0. Suppose for a

contradiction that x−1 ≤ 0. Then x−1 < 0. Now x > 0 implies that 1 = x·x−1 ≤ 0·x−1 = 0, contradiction. Now let 0 < x ≤ y. We know that x−1,y−1 > 0. Therefore x−1 = x−1 ·1 = x−1yy−1 ≥ x−1xy−1 = y−1.

• Theorem. Let (F,+,·) be a totally ordered field let x ∈ F.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Theorem. Let (F,+,·) be a totally ordered field and let x,y ∈ F

be so that 0 < x ≤ y. Then 0 < y−1 ≤ x−1.

• Proof. We first prove that x > 0 implies x−1 > 0. Suppose for a

contradiction that x−1 ≤ 0. Then x−1 < 0. Now x > 0 implies that 1 = x·x−1 ≤ 0·x−1 = 0, contradiction. Now let 0 < x ≤ y. We know that x−1,y−1 > 0. Therefore x−1 = x−1 ·1 = x−1yy−1 ≥ x−1xy−1 = y−1.

• Theorem. Let (F,+,·) be a totally ordered field let x ∈ F. Then

x2 ≥ 0.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Theorem. Let (F,+,·) be a totally ordered field and let x,y ∈ F

be so that 0 < x ≤ y. Then 0 < y−1 ≤ x−1.

• Proof. We first prove that x > 0 implies x−1 > 0. Suppose for a

contradiction that x−1 ≤ 0. Then x−1 < 0. Now x > 0 implies that 1 = x·x−1 ≤ 0·x−1 = 0, contradiction. Now let 0 < x ≤ y. We know that x−1,y−1 > 0. Therefore x−1 = x−1 ·1 = x−1yy−1 ≥ x−1xy−1 = y−1.

• Theorem. Let (F,+,·) be a totally ordered field let x ∈ F. Then

x2 ≥ 0. Proof.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Theorem. Let (F,+,·) be a totally ordered field and let x,y ∈ F

be so that 0 < x ≤ y. Then 0 < y−1 ≤ x−1.

• Proof. We first prove that x > 0 implies x−1 > 0. Suppose for a

contradiction that x−1 ≤ 0. Then x−1 < 0. Now x > 0 implies that 1 = x·x−1 ≤ 0·x−1 = 0, contradiction. Now let 0 < x ≤ y. We know that x−1,y−1 > 0. Therefore x−1 = x−1 ·1 = x−1yy−1 ≥ x−1xy−1 = y−1.

• Theorem. Let (F,+,·) be a totally ordered field let x ∈ F. Then

x2 ≥ 0.

• Proof. Exercise.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields

logo1 Positive Rational Numbers Ordered Fields

• Theorem. Let (F,+,·) be a totally ordered field and let x,y ∈ F

be so that 0 < x ≤ y. Then 0 < y−1 ≤ x−1.

• Proof. We first prove that x > 0 implies x−1 > 0. Suppose for a

contradiction that x−1 ≤ 0. Then x−1 < 0. Now x > 0 implies that 1 = x·x−1 ≤ 0·x−1 = 0, contradiction. Now let 0 < x ≤ y. We know that x−1,y−1 > 0. Therefore x−1 = x−1 ·1 = x−1yy−1 ≥ x−1xy−1 = y−1.

• Theorem. Let (F,+,·) be a totally ordered field let x ∈ F. Then

x2 ≥ 0.

• Proof. Exercise.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Ordered Fields