Explanation Given information: U n is a set of all n th roots of 1 . Formula used: 1) Definition of group: Suppose the binary operation ∗ is defined for the element of set G . G is a group with respect to ∗ , provided the following conditions hold. 1. G is closed under ∗ , that is x ∈ G and y ∈ G implies that x ∗ y is in G 2. ∗ is associative. For all x , y , z in G , x ∗ ( y ∗ z ) = ( x ∗ y ) ∗ z 3. G has an identity element e . There is an e in G , such that x ∗ e = e ∗ x = x for all x ∈ G 4. G contains inverses. For each a ∈ G , there exists b ∈ G , such that a ∗ b = b ∗ a = e 2) Theorem: Product of the Complex Numbers: If z 1 = r 1 ( cos θ 1 + i sin θ 1 ) and are arbitrary complex numbers in trigonometric form, then z 1 z 2 = r 1 r 2 [ cos ( θ 1 + θ 2 ) + i sin ( θ 1 + θ 2 ) ] . Proof: Let U n is a set of all n th roots of 1 . Then, the set is given by, U n = { cos 2 k π n + i sin 2 k π n | 0 ≤ k ≤ n − 1 } Now, U n forms a group if it satisfies the following conditions: (i) Let z 1 , z 2 ∈ U n such that z 1 = cos 2 k 1 π n + i sin 2 k 1 π n and z 1 = cos 2 k 1 π n + i sin 2 k 1 π n Therefore, z 1 ⋅ z 2 = ( cos 2 k 1 π n + i sin 2 k 1 π n ) ( cos 2 k 2 π n + i sin 2 k 2 π n ) By using the theorem, = cos 2 k 1 π + 2 k 2 π n + i sin 2 k 1 π + 2 k 2 π n = cos 2 π ( k 1 + k 2 ) n + i sin 2 π ( k 1 + k 2 ) n Let k 1 + k 2 = r , 0 ≤ r < n ⇒ z 1 ⋅ z 2 = cos 2 π r n + i sin 2 π r n ∈ U n Therefore, U n is closed under multiplication. (ii) As U n is a subset of the set of complex numbers, that is ℂ and multiplication in ℂ is associative, then multiplication in U n is also associative. (iii) Let z ∈ U n , then z = cos 2 k π n + i sin 2 k π n , 0 ≤ k ≤ n − 1

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Author: Gilbert, Linda, Jimmie

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Chapter 7.3, Problem 10E

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To prove: For a fixed value of n, the set Un of all n th roots of 1 forms a group with respect to multiplication.

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Chapter 7.3, Problem 9E

Chapter 7.3, Problem 11E

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Chapter 7 Solutions

Elements Of Modern Algebra

Ch. 7.1 - Prob. 2E Ch. 7.1 - Prob. 3E Ch. 7.1 - Find the decimal representation for each of the...Ch. 7.1 - Prob. 5E Ch. 7.1 - Prob. 6E Ch. 7.1 - Prob. 7E Ch. 7.1 - Prob. 8E Ch. 7.1 - Express each of the numbers in Exercises 7-12 as a...Ch. 7.1 - Express each of the numbers in Exercises 7-12 as a...Ch. 7.1 - Express each of the numbers in Exercises 7-12 as a...Ch. 7.1 - Express each of the numbers in Exercises 7-12 as a...Ch. 7.1 - Prove that is irrational. (That is, prove there...Ch. 7.1 - Prove that is irrational. Ch. 7.1 - Prove that if is a prime integer, then is...Ch. 7.1 - Prove that if a is rational and b is irrational,...Ch. 7.1 - Prove that if is a nonzero rational number and ...Ch. 7.1 - Prove that if is an irrational number, then is...Ch. 7.1 - Prove that if is a nonzero rational number and ...Ch. 7.1 - Give counterexamples for the following...Ch. 7.1 - Let S be a nonempty subset of an order field F....Ch. 7.1 - Prove that if F is an ordered field with F+ as its...Ch. 7.1 - If F is an ordered field, prove that F contains a...Ch. 7.1 - Prove that any ordered field must contain a...Ch. 7.1 - If and are positive real numbers, prove that...Ch. 7.1 - Prove that if and are real numbers such that ,...Ch. 7.2 - True or False Label each of the following...Ch. 7.2 - Prob. 2TFE Ch. 7.2 - Prob. 3TFE Ch. 7.2 - True or False Label each of the following...Ch. 7.2 - Prob. 5TFE Ch. 7.2 - True or False Label each of the following...Ch. 7.2 - Prob. 7TFE Ch. 7.2 - Prob. 1E Ch. 7.2 - Prob. 2E Ch. 7.2 - Prob. 3E Ch. 7.2 - Prob. 4E Ch. 7.2 - Prob. 5E Ch. 7.2 - Prob. 6E Ch. 7.2 - Prob. 7E Ch. 7.2 - Prob. 8E Ch. 7.2 - Prob. 9E Ch. 7.2 - Prob. 10E Ch. 7.2 - Prob. 11E Ch. 7.2 - Prob. 12E Ch. 7.2 - Prob. 13E Ch. 7.2 - Prob. 14E Ch. 7.2 - Prob. 15E Ch. 7.2 - Prob. 16E Ch. 7.2 - Prob. 17E Ch. 7.2 - Prob. 18E Ch. 7.2 - Prob. 19E Ch. 7.2 - Prob. 20E Ch. 7.2 - Prob. 21E Ch. 7.2 - Prob. 22E Ch. 7.2 - Prob. 23E Ch. 7.2 - Prob. 24E Ch. 7.2 - Prob. 25E Ch. 7.2 - Prob. 26E Ch. 7.2 - Prob. 27E Ch. 7.2 - Prob. 28E Ch. 7.2 - Prob. 29E Ch. 7.2 - Prob. 30E Ch. 7.2 - Prob. 31E Ch. 7.2 - Prob. 32E Ch. 7.2 - Prob. 33E Ch. 7.2 - Prob. 34E Ch. 7.2 - Prob. 35E Ch. 7.2 - Prob. 36E Ch. 7.2 - Prob. 37E Ch. 7.2 - Prob. 38E Ch. 7.2 - Prob. 39E Ch. 7.2 - Prob. 40E Ch. 7.2 - Exercise are stated using the notation in the...Ch. 7.2 - Prob. 42E Ch. 7.2 - Prob. 43E Ch. 7.2 - Prob. 44E Ch. 7.2 - Prob. 45E Ch. 7.2 - Prob. 46E Ch. 7.2 - Prob. 47E Ch. 7.2 - Prob. 48E Ch. 7.2 - Prob. 49E Ch. 7.2 - Prob. 50E Ch. 7.2 - An element in a ring is idempotent if . Prove...Ch. 7.2 - Prove that a finite ring R with unity and no zero...Ch. 7.3 - True or False Label each of the following...Ch. 7.3 - Prob. 2TFE Ch. 7.3 - Prob. 3TFE Ch. 7.3 - Prob. 4TFE Ch. 7.3 - Prob. 1E Ch. 7.3 - Find each of the following products. Write each...Ch. 7.3 - Prob. 3E Ch. 7.3 - Show that the n distinct n th roots of 1 are...Ch. 7.3 - Prob. 5E Ch. 7.3 - Prob. 6E Ch. 7.3 - Prob. 7E Ch. 7.3 - Prob. 8E Ch. 7.3 - Prob. 9E Ch. 7.3 - Prob. 10E Ch. 7.3 - Prob. 11E Ch. 7.3 - Prob. 12E Ch. 7.3 - Prob. 13E Ch. 7.3 - Prob. 14E Ch. 7.3 - Prove that the group in Exercise is cyclic, with ...Ch. 7.3 - Prob. 16E Ch. 7.3 - Prob. 17E Ch. 7.3 - Prob. 18E Ch. 7.3 - Prob. 19E Ch. 7.3 - Prob. 20E Ch. 7.3 - Prob. 21E Ch. 7.3 - Prob. 22E Ch. 7.3 - Prove that the set of all complex numbers that...Ch. 7.3 - Prob. 24E Ch. 7.3 - Prob. 25E Ch. 7.3 - Prob. 26E Ch. 7.3 - Prob. 27E Ch. 7.3 - Prob. 28E

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To prove: For a fixed value of n , the set U n of all n th roots of 1 forms a group with respect to multiplication.

Solution Summary: The author explains that U_n forms a group with respect to multiplication for 1, provided the following conditions hold.

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To prove: For a fixed value of n, the set Un of all n th roots of 1 forms a group with respect to multiplication.

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Chapter 7 Solutions