Consider the set S = {1, ω, ω2}, where ω and ω2 are cube roots of unity. If *…
2010
Consider the set S = {1, ω, ω2}, where ω and ω2 are cube roots of unity. If * denotes the multiplication operation, the structure (S, *) forms
- A.
A group
- B.
A ring
- C.
An integral domain
- D.
A field
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Correct answer: A
Key idea: verify the group axioms for (S, *) where S = {1, ω, ω2} and ω3 = 1.
Closure: Multiplying any two elements from S yields one of 1, ω, ω2 (use ω·ω = ω2, ω·ω2 = ω3 = 1, etc.).
Associativity: Inherited from associative multiplication of complex numbers.
Identity: 1 acts as the multiplicative identity.
Inverses: Each element has an inverse in S (1^{-1}=1, ω^{-1}=ω2, ω2^{-1}=ω).
Conclusion: These facts show (S, *) is a group. It is cyclic of order 3 (generated by ω) and abelian.