Abstract Algebra Theory and Applications - Computer Science ...

64 CHAPTER 3 CYCLIC GROUPSTheorem 3.11 If z n = 1, then the nth roots of unity are( ) 2kπz = cis ,nwhere k = 0, 1, . . . , n − 1. Furthermore, the nth roots of unity form a cyclicsubgroup of T of order n.Proof. By DeMoivre’s Theorem,(z n = cis n 2kπ )= cis(2kπ) = 1.nThe z’s are distinct since the numbers 2kπ/n are all distinct and are greaterthan or equal to 0 but less than 2π. The fact that these are all of the rootsof the equation z n = 1 follows from the Fundamental Theorem of Algebra(Theorem 19.16), which states that a polynomial of degree n can have atmost n roots. We will leave the proof that the nth roots of unity form acyclic subgroup of T as an exercise.□A generator for the group of the nth roots of unity is called a primitiventh root of unity.Example 11. The 8th roots of unity can be represented as eight equallyspaced points on the unit circle (Figure 3.4). The primitive 8th roots ofunity are√ √2 2ω =2 + 2 i√ √2 2ω 3 = −2 + 2 i√ √2 2ω 5 = −2 − 2 i√ √2 2ω 7 =2 − 2 i. 3.3 The Method of Repeated Squares 11 The results in this section are needed only in Chapter 6.