Consider the set S = {1, omega, omega^2}, where omega and omega^2 are cube roots of unity. If * denotes the multiplicati...

Question

Consider the set S = {1, omega, omega^2}, where omega and omega^2 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

Accepted Answer

All group axioms hold under the single operation, so (S, *) is a Group; answer A. We have only one operation here, multiplication, acting on the three cube roots of unity {1, omega, omega^2}. A ring, integral domain, and field all need TWO operations (addition and multiplication), so they are ruled out immediately. With a single operation that is closed, associative, has identity 1, and gives every element an inverse, the structure is a group.

Consider the set S = {1, omega, omega^2}, where omega and omega^2 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