Read through the following theorem and proof and answer the questions on it that follow. Be sure to number your responses clearly and correctly. Theorem: Let F be an ordered field, and let a, b e F with a, b > 0. If a² < b² then a < b. Proof: (1) Suppose that a? < b², i.e. a² – b2 < 0 where a, b > 0. (2) This means that (a + b)(a – b) < 0. (3) From this we get that (a + b)-1. [(a+ b)(a – b)] < (a + b)-1 . 0. (4) Hence a – b < 0, i.e. a < b. QED - 1.1 Explain why the inverse (a + b)-1 of a + b exists. 1.2 Name the three field properties needed that one makes use of in reducing the expression (a + b)-1 . [(a + b)(a – b)] on the LHS of (3) to a – 6.
Read through the following theorem and proof and answer the questions on it that follow. Be sure to number your responses clearly and correctly. Theorem: Let F be an ordered field, and let a, b e F with a, b > 0. If a² < b² then a < b. Proof: (1) Suppose that a? < b², i.e. a² – b2 < 0 where a, b > 0. (2) This means that (a + b)(a – b) < 0. (3) From this we get that (a + b)-1. [(a+ b)(a – b)] < (a + b)-1 . 0. (4) Hence a – b < 0, i.e. a < b. QED - 1.1 Explain why the inverse (a + b)-1 of a + b exists. 1.2 Name the three field properties needed that one makes use of in reducing the expression (a + b)-1 . [(a + b)(a – b)] on the LHS of (3) to a – 6.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter5: Rings, Integral Domains, And Fields
Section5.4: Ordered Integral Domains
Problem 14E: 14. a. If is an ordered integral domain, prove that each element in the quotient field of ...
Related questions
Question
![Read through the following theorem and proof and answer the questions on
it that follow. Be sure to number your responses clearly and correctly.
Theorem: Let F be an ordered field, and let a, b e F with a, b > 0. If a² < b²
then a < b.
Proof: (1) Suppose that a? < b², i.e. a² – b2 < 0 where a, b > 0.
(2) This means that (a + b)(a – b) < 0.
(3) From this we get that (a + b)-1. [(a+ b)(a – b)] < (a + b)-1 . 0.
(4) Hence a – b < 0, i.e. a < b. QED
-
1.1 Explain why the inverse (a + b)-1 of a + b exists.
1.2 Name the three field properties needed that one makes use of in reducing
the expression (a + b)-1 . [(a + b)(a – b)] on the LHS of (3) to a – 6.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdead8ce3-beeb-4259-a768-228590e4ae28%2F27671ca1-22aa-4ee0-8782-aee66f7e59bd%2Fsqo8px_processed.png&w=3840&q=75)
Transcribed Image Text:Read through the following theorem and proof and answer the questions on
it that follow. Be sure to number your responses clearly and correctly.
Theorem: Let F be an ordered field, and let a, b e F with a, b > 0. If a² < b²
then a < b.
Proof: (1) Suppose that a? < b², i.e. a² – b2 < 0 where a, b > 0.
(2) This means that (a + b)(a – b) < 0.
(3) From this we get that (a + b)-1. [(a+ b)(a – b)] < (a + b)-1 . 0.
(4) Hence a – b < 0, i.e. a < b. QED
-
1.1 Explain why the inverse (a + b)-1 of a + b exists.
1.2 Name the three field properties needed that one makes use of in reducing
the expression (a + b)-1 . [(a + b)(a – b)] on the LHS of (3) to a – 6.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage