Notes on the Complex Roots of Unity

Roots of unity are the complex solutions of z^n = 1, compactly given by z_k = e^{2π i k/n} for k = 0,...,n−1. Representing unit-magnitude complex numbers in polar form makes these solutions immediately visible as n equally spaced points on the unit circle; simple examples are n=3 (1, −1/2 ± i√3/2) and n=4 (1, −1, i, −i). Connecting adjacent roots produces regular n-gons, multiplying roots corresponds to rotating around the circle, and the inverse (or complex conjugate) of z_k is z_{n−k}. The set μ_n = {1,w,w^2,...,w^{n−1}} with w = e^{2π i/n} is a finite abelian group under multiplication and is cyclic: w generates μ_n, and any w^k is a generator exactly when gcd(k,n)=1 (those generators are primitive roots).

Summing the n roots yields 0 because they form a symmetric geometric progression; more generally ∑_{i=0}^{n−1} w^{ki} equals n if k ≡ 0 (mod n) and 0 otherwise. Treating the vectors of root powers v_j = (1, w^j, w^{2j},...,w^{(n−1)j}) produces an n×n Fourier matrix V whose rows (and columns) are mutually orthogonal: ⟨v_i,v_j⟩ = n when i=j and 0 otherwise. Those n orthogonal vectors form a basis of C^n, underpinning the Discrete Fourier Transform. Counting multiplicity, z^n−1 has exactly n complex roots by the Fundamental Theorem of Algebra.