Abstract Algebra Theory and Applications - Computer Science ...

3.2 THE GROUP C ∗ 63Proof. We will use induction on n. For n = 1 the theorem is trivial.Assume that the theorem is true for all k such that 1 ≤ k ≤ n. Thenz n+1 = z n z= r n (cos nθ + i sin nθ)r(cos θ + i sin θ)= r n+1 [(cos nθ cos θ − sin nθ sin θ) + i(sin nθ cos θ + cos nθ sin θ)]= r n+1 [cos(nθ + θ) + i sin(nθ + θ)]= r n+1 [cos(n + 1)θ + i sin(n + 1)θ].Example 10. Suppose that z = 1 + i and we wish to compute z 10 . Ratherthan computing (1 + i) 10 directly, it is much easier to switch to polar coordinatesand calculate z 10 using DeMoivre’s Theorem:z 10 = (1 + i) 10(√ ( π)) 10= 2 cis4= ( √ ( ) 5π2 ) 10 cis2( π)= 32 cis2= 32i.The Circle Group and the Roots of UnityThe multiplicative group of the complex numbers, C ∗ , possesses some interestingsubgroups. Whereas Q ∗ and R ∗ have no interesting subgroups offinite order, C ∗ has many. We first consider the circle group,T = {z ∈ C : |z| = 1}.The following proposition is a direct result of Proposition 3.8.Proposition 3.10 The circle group is a subgroup of C ∗ .Although the circle group has infinite order, it has many interesting finitesubgroups. Suppose that H = {1, −1, i, −i}. Then H is a subgroup of thecircle group. Also, 1, −1, i, and −i are exactly those complex numbers thatsatisfy the equation z 4 = 1. The complex numbers satisfying the equationz n = 1 are called the nth roots of unity.□