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Appendix C: Fields Problem 1. Consider the following set of complex numbers Q(i) = (a + bi: a, b € Q}. Prove that Q(i) is a field under the usual addition and multiplication operations.

Appendix C: Fields Problem 1. Consider the following set of complex numbers Q(i) = (a + bi: a, b € Q}. Prove that Q(i) is a field under the usual addition and multiplication operations.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Appendix C: Fields Problem 1. Consider the following set of complex numbers Q(i) = (a + bi: a, b e Q}. Prove that Q(i) is a field under the usual addition and multiplication operations. Problem. Let n be a positive integer and define Q(√n) = {a+b√n: a, b = Q}. Prove that Q(√n) is a field under the usual addition and multiplication operations. Note that QC Q(i) C C and similarly QC Q(√n) c R. Whenever a field F is contained into another field K, we say that F is a subfield of K. For example, we say that R is a subfield of C. Moreover, Q(i) is a subfield of C and Q(√2) is a subfield of R. 1 Vector spaces 1.2 Vector spaces Problem 2. Let V = {(a₁, a2): a₁, a2 € R}. For (a₁, a2), (b₁,b2) € V and c ER, define (a₁, a2) + (b₁,b₂) = (a₁ +2b₁, a2 + 3b₂) and c(a₁, a2) = = (ca₁, ca₂). Is V a vector space over R with these operations? Justify your answer. 1
Transcribed Image Text:Appendix C: Fields Problem 1. Consider the following set of complex numbers Q(i) = (a + bi: a, b e Q}. Prove that Q(i) is a field under the usual addition and multiplication operations. Problem. Let n be a positive integer and define Q(√n) = {a+b√n: a, b = Q}. Prove that Q(√n) is a field under the usual addition and multiplication operations. Note that QC Q(i) C C and similarly QC Q(√n) c R. Whenever a field F is contained into another field K, we say that F is a subfield of K. For example, we say that R is a subfield of C. Moreover, Q(i) is a subfield of C and Q(√2) is a subfield of R. 1 Vector spaces 1.2 Vector spaces Problem 2. Let V = {(a₁, a2): a₁, a2 € R}. For (a₁, a2), (b₁,b2) € V and c ER, define (a₁, a2) + (b₁,b₂) = (a₁ +2b₁, a2 + 3b₂) and c(a₁, a2) = = (ca₁, ca₂). Is V a vector space over R with these operations? Justify your answer. 1
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