Another FieldWe'll let Q(√2) represent the set of numbers that can be written in theform a+b√2, where 'a' and 'b' and rational numbers (i.e., 'a' and 'b' are from132the set Q). For example, +¹3√2, 7.2-6√2, and 15 (= 15 + 0√2) are threenumbers that would be in the set Q(√2), but √√3+ 4√2 or 3 + √2 would notbe in the set Q(√2) (neither √√3 nor are rational numbers). A number such3+5√2 would be in the set Q(√2) since 3+√2 can be re-written in the form7a+b√2 using the following steps:773-5√2 21-35√221-35√2 21 - 35√2 21 35== + √23+5√2 3+5√2 3-5√2 9 - 25(2) -4141 41=*Your task: Show that the system (Q(√2), +, *) is a field, where + is theordinary addition operation and * is the ordinary multiplication operation.So, you need to show that:a. (Q(√2), +, *) is a commutative ring, andb. (Q(√2) - {0}, *) is a group.

Given: A set Q2,+,* which is defined as Q2=a+b2 | a,b∈Q. To Prove: (a) Q2,+,* is a commutative…

Another Field 13 We'll let Q(√2) represent the set of numbers that can be written in the form a+b√2, where 'a' and 'b' and rational numbers (i.e., 'a' and 'b' are from the set Q). For example, +¹3³√2, 7.2 – 6√2, and 15 (= 15 + 0√2) are three numbers that would be in the set Q(√2), but √√3+ 4√2 or 3 + √2 would not be in the set Q(√2) (neither √3 nor are rational numbers). A number such as 3+√ would be in the set Q(√2) since - can be re-written in the form a+b√2 using the following steps: 3+5√2 7 7 3-5√2 21-35√2 = 3+5√2 3+5√2 3-5√2 9 - 25(2) * 21-35√2 -41 a. (Q(√2), +, *) is a commutative ring, and b. (Q(√2) - {0}, *) is a group. 21 35 41 41 √2 Your task: Show that the system (Q(√2), +, *) is a field, where + is the ordinary addition operation and is the ordinary multiplication operation. So, you need to show that:

Another Field 13 We'll let Q(√2) represent the set of numbers that can be written in the form a+b√2, where 'a' and 'b' and rational numbers (i.e., 'a' and 'b' are from the set Q). For example, +¹3³√2, 7.2 – 6√2, and 15 (= 15 + 0√2) are three numbers that would be in the set Q(√2), but √√3+ 4√2 or 3 + √2 would not be in the set Q(√2) (neither √3 nor are rational numbers). A number such as 3+√ would be in the set Q(√2) since - can be re-written in the form a+b√2 using the following steps: 3+5√2 7 7 3-5√2 21-35√2 = 3+5√2 3+5√2 3-5√2 9 - 25(2) * 21-35√2 -41 a. (Q(√2), +, ) is a commutative ring, and b. (Q(√2) - {0}, ) is a group. 21 35 41 41 √2 Your task: Show that the system (Q(√2), +, *) is a field, where + is the ordinary addition operation and is the ordinary multiplication operation. So, you need to show that:

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter1: Fundamental Concepts Of Algebra

Section1.2: Exponents And Radicals

Problem 91E

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