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B All Ⓒ (i), (iii) D None E (i), (ii), (iv) (ii), (iii) Which of the following are true ? (i): R= {a+b√3|a, b € Z}. Then (R, +,-) is an integral domain. (ii): R= {a+b√ a, b e Z}. Then (R, +,) is an integral domain. (iii): R = {a+b√3 | a, b = Q}. Then (R, +,-) is a field. (iv): R= {a+b√3|a, b € Z}. Then (R, +,-) is a field. ...

B All Ⓒ (i), (iii) D None E (i), (ii), (iv) (ii), (iii) Which of the following are true ? (i): R= {a+b√3|a, b € Z}. Then (R, +,-) is an integral domain. (ii): R= {a+b√ a, b e Z}. Then (R, +,) is an integral domain. (iii): R = {a+b√3 | a, b = Q}. Then (R, +,-) is a field. (iv): R= {a+b√3|a, b € Z}. Then (R, +,-) is a field. ...

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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The question asks which of the following statements are true: (i) \( R = \{ a + b \sqrt{3} \mid a, b \in \mathbb{Z} \} \). Then \( (R, +, \cdot) \) is an integral domain. (ii) \( R = \{ a + b \sqrt{\pi} \mid a, b \in \mathbb{Z} \} \). Then \( (R, +, \cdot) \) is an integral domain. (iii) \( R = \{ a + b \sqrt{3} \mid a, b \in \mathbb{Q} \} \). Then \( (R, +, \cdot) \) is a field. (iv) \( R = \{ a + b \sqrt{3} \mid a, b \in \mathbb{Z} \} \). Then \( (R, +, \cdot) \) is a field. Options for answers are: A) None B) All C) (i), (iii) D) (i), (ii), (iv) E) (ii), (iii)
Transcribed Image Text:The question asks which of the following statements are true: (i) \( R = \{ a + b \sqrt{3} \mid a, b \in \mathbb{Z} \} \). Then \( (R, +, \cdot) \) is an integral domain. (ii) \( R = \{ a + b \sqrt{\pi} \mid a, b \in \mathbb{Z} \} \). Then \( (R, +, \cdot) \) is an integral domain. (iii) \( R = \{ a + b \sqrt{3} \mid a, b \in \mathbb{Q} \} \). Then \( (R, +, \cdot) \) is a field. (iv) \( R = \{ a + b \sqrt{3} \mid a, b \in \mathbb{Z} \} \). Then \( (R, +, \cdot) \) is a field. Options for answers are: A) None B) All C) (i), (iii) D) (i), (ii), (iv) E) (ii), (iii)
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