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(a) Prove that the set O equipped with the addition and multiplication of complex num- bers satisfies the following properties: (i) (ii) (iii) is closed under addition: for any a, ß € 0, we have a + B € 0. is closed under negation: for any a € 0, we have -a € 0. is closed under multiplication: for any a, ß = 0, we have aß € 0. Remark. In the terms of Algebra, is a subring of the ring C of complex numbers. (b) Consider the integer-valued function N defined on 0: N(a+b√-5) = a² +56². Prove that N(aß) = N(a)N (B) for any two elements a and 3 in 0. Remark. Say that an element a € O divides another element 3 € O, denoted by a | B if there is an elementy E O such that = ay. Hence, problem 1.(b) shows that a | B⇒ N(a) | N(B). (c) Say that an element & EO is a unit if ɛ divides 1. Prove that all the units in are 1 and -1. Hint. Assume & E O is a unit other than ±1, then use problem 1.(b).

(a) Prove that the set O equipped with the addition and multiplication of complex num- bers satisfies the following properties: (i) (ii) (iii) is closed under addition: for any a, ß € 0, we have a + B € 0. is closed under negation: for any a € 0, we have -a € 0. is closed under multiplication: for any a, ß = 0, we have aß € 0. Remark. In the terms of Algebra, is a subring of the ring C of complex numbers. (b) Consider the integer-valued function N defined on 0: N(a+b√-5) = a² +56². Prove that N(aß) = N(a)N (B) for any two elements a and 3 in 0. Remark. Say that an element a € O divides another element 3 € O, denoted by a | B if there is an elementy E O such that = ay. Hence, problem 1.(b) shows that a | B⇒ N(a) | N(B). (c) Say that an element & EO is a unit if ɛ divides 1. Prove that all the units in are 1 and -1. Hint. Assume & E O is a unit other than ±1, then use problem 1.(b).

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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Problem 1. In chapter 4 of the textbook, we see that Gaussian integers and Eisenstein integers also have unique prime factorization. However, this property is not always satisfied. The following problems lead to a counterexample. Let's consider the collection of complex numbers of the form 0 := {a+b√−5 | a,b ≤Z}.
Transcribed Image Text:Problem 1. In chapter 4 of the textbook, we see that Gaussian integers and Eisenstein integers also have unique prime factorization. However, this property is not always satisfied. The following problems lead to a counterexample. Let's consider the collection of complex numbers of the form 0 := {a+b√−5 | a,b ≤Z}.
(a) Prove that the set O equipped with the addition and multiplication of complex num- bers satisfies the following properties: (i) is closed under addition: for any a, ß € 0, we have a + ß € 0. (ii) is closed under negation: for any a € 0, we have -a € 0. (iii) Ø is closed under multiplication: for any a, ß € 0, we have aß ≤ 0. Remark. In the terms of Algebra, is a subring of the ring C of complex numbers. (b) Consider the integer-valued function N defined on 0: N(a+b√-5) := a² +56². Prove that N(aß) = N(a)N(B) for any two elements a and B in 0. Remark. Say that an element a € O divides another element 3 € O, denoted by a | B if there is an element y E O such that = ay. Hence, problem 1.(b) shows that a | B ⇒N(a) | N (B). (c) Say that an element & E O is a unit if e divides 1. Prove that all the units in are 1 and -1. Hint. Assume & € O is a unit other than ±1, then use problem 1.(b).
Transcribed Image Text:(a) Prove that the set O equipped with the addition and multiplication of complex num- bers satisfies the following properties: (i) is closed under addition: for any a, ß € 0, we have a + ß € 0. (ii) is closed under negation: for any a € 0, we have -a € 0. (iii) Ø is closed under multiplication: for any a, ß € 0, we have aß ≤ 0. Remark. In the terms of Algebra, is a subring of the ring C of complex numbers. (b) Consider the integer-valued function N defined on 0: N(a+b√-5) := a² +56². Prove that N(aß) = N(a)N(B) for any two elements a and B in 0. Remark. Say that an element a € O divides another element 3 € O, denoted by a | B if there is an element y E O such that = ay. Hence, problem 1.(b) shows that a | B ⇒N(a) | N (B). (c) Say that an element & E O is a unit if e divides 1. Prove that all the units in are 1 and -1. Hint. Assume & € O is a unit other than ±1, then use problem 1.(b).
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