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B E None (i), (ii) (iii), (iv) Which of the following are ring homomorphisms? (i): f: (Q(√2), +,-) → (Q(√3), +,-), a +b√2+a+b√3. (ii): f: C Rx R, a +bi (a, b). (iii): f: (QxQ, +,-) → (Q(√2), +,-), (x,y) →x+yv2. (iv): f: CR, a +bi+ a² + b². ...

B E None (i), (ii) (iii), (iv) Which of the following are ring homomorphisms? (i): f: (Q(√2), +,-) → (Q(√3), +,-), a +b√2+a+b√3. (ii): f: C Rx R, a +bi (a, b). (iii): f: (QxQ, +,-) → (Q(√2), +,-), (x,y) →x+yv2. (iv): f: CR, a +bi+ a² + b². ...

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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**Question:** Which of the following are ring homomorphisms? **Options:** (i) \( f : (\mathbb{Q}(\sqrt{2}), +, \cdot) \to (\mathbb{Q}(\sqrt{3}), +, \cdot) \), defined by \( a + b\sqrt{2} \mapsto a + b\sqrt{3} \). (ii) \( f : \mathbb{C} \to \mathbb{R} \times \mathbb{R}, \, a + bi \mapsto (a, b) \). (iii) \( f : (\mathbb{Q} \times \mathbb{Q}, +, \cdot) \to (\mathbb{Q}(\sqrt{2}), +, \cdot) \), defined by \( (x, y) \mapsto x + y\sqrt{2} \). (iv) \( f : \mathbb{C} \to \mathbb{R}, \, a + bi \mapsto a^2 + b^2 \). **Choices:** - A: None - B: (i) - C: (i), (ii) - D: (ii) - E: (iii), (iv)
Transcribed Image Text:**Question:** Which of the following are ring homomorphisms? **Options:** (i) \( f : (\mathbb{Q}(\sqrt{2}), +, \cdot) \to (\mathbb{Q}(\sqrt{3}), +, \cdot) \), defined by \( a + b\sqrt{2} \mapsto a + b\sqrt{3} \). (ii) \( f : \mathbb{C} \to \mathbb{R} \times \mathbb{R}, \, a + bi \mapsto (a, b) \). (iii) \( f : (\mathbb{Q} \times \mathbb{Q}, +, \cdot) \to (\mathbb{Q}(\sqrt{2}), +, \cdot) \), defined by \( (x, y) \mapsto x + y\sqrt{2} \). (iv) \( f : \mathbb{C} \to \mathbb{R}, \, a + bi \mapsto a^2 + b^2 \). **Choices:** - A: None - B: (i) - C: (i), (ii) - D: (ii) - E: (iii), (iv)
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