The n-th roots of unity

Let n > 1 be an integer. An nth root of unity in ℂ is a complex number α which is a zero of xⁿ − 1. The set of nth roots of unity form a cyclic group of order n, 𝐶_n, under multiplication and each generator of 𝐶_n is called a primitive nth root of unity.
Any such α is of the form α_k = e^2ik/n, where 1 ≤ k < n or equivalently

α_{k} = \cos \frac{2 k π}{n} + i \sin \frac{2 k π}{n}, k = 0, 1, 2, \dots, n - 1

Let G = {α₀, α₁, ..., α_n}. We have

α_{k} = \cos \frac{2 k π}{n} + i \sin \frac{2 k π}{n} = {(\cos \frac{2 π}{n} + i \sin \frac{2 π}{n})}^{k} = α_{1}^{k} k = 0, 1, 2, \dots, n - 1

Hence we find that G = ⟨α₁⟩, and so, G is a cyclic group of order n.

Definition 7.3.1. If α is an nth root of unity and n is the smallest positive integer for which αⁿ = 1, we say that α is a primitive nth root of unity. □

So a nth primitive root of unit is an element of the group with order exactly n.

For example, i is an 8th root of unity (for i⁸ = 1), but not a primitive 8th root of unity; i is a primitive 4th root of unity.

Now for any integer 1 ≤ k < n, α₁^k is a generator of G iff gcd(k,n) = 1, see Lemma 7.3.10. If n = 8 we have, α₁¹, α₁³, α₁⁵ and α₁⁷ are generators of this cyclic group. The term "primitive" exactly refers to being a generator of the cyclic group, namely an nth root of unity is primitive when there is no positive integer k smaller than n such that α_n^k = 1. ■

7.3.2 Proposition. The set of n-th roots of unity in ℂ forms a cyclic group 𝐶_n isomorphic to (ℤ/nℤ,+).

Proof. Consider the group homomorphism f: (ℤ, +) ⟶ ℂ* defined by f(k) = e^2ikπ/n. The kernel of this homomorphism is exactly nℤ, and its image 𝐶_n, which gives the result. (We could just note that both groups are cyclic and both have order n, so by general principles they are isomorphic.) □

The ℤ/nℤ group

The elements of ℤ/nℤ form an abelian group under addition which is isomorphic to s_n the set {1, α₁, ..., α₁^{n − 1}}, under multiplication. The special case n = 2 gives 𝐶₂ as the group {1, −1}, under mutliplication. In general, the group elements lie on the unit circle in the plane ℂ, and multiplication by α₁^m really rotates the elements through m places. Imagine the congruence classes nℤ, nℤ + 1, ..., nℤ + n − 1 (equivalently [0], [1],..., [n−1]), arranged in a circle, then addition of nℤ + m in the same way seen for 𝐶_n, corresponds to rotation through m places.

5th roots unity — Elements of 𝐶₅. α₁ is the generator of the group.

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