Roots of Unity: Properties, Sums and Applications Guide

The roots of unity are the n complex solutions of z^n = 1, spaced evenly around the unit circle as the vertices of a regular n-gon: ω_k = cos(2πk/n) + i sin(2πk/n) = e^{2πik/n} for k = 0, 1, up to n − 1.

Their symmetry forces a striking cancellation: for n > 1 the sum of all n roots is exactly zero. That vanishing sum powers the discrete Fourier transform, digital signal processing, and combinatorial "root-of-unity filters" that select terms by divisibility.

Key Concepts

• n-th roots of unity: the n distinct numbers e^{2πik/n}; products of roots are roots, so they form a group.
• Primitive root: ω with ω^m ≠ 1 for 1 ≤ m < n, i.e. ω_k with k coprime to n; for n = 4, i is primitive but −1 is not.
• Sum identity: 1 + ω + ω² + … + ω^{n−1} = 0 for n > 1 with ω = e^{2πi/n}.
• Product identity: the product of all n roots is (−1)^{n+1}: 1 for odd n, −1 for even n.
• Root-of-unity filter: (1/n)Σ_{k=0}^{n−1} ω^{km} equals 1 if n divides m, else 0.

Formulas

• ω_k = e^{2πik/n}, k = 0, …, n − 1.
• Σ_{k=0}^{n−1} (ω^j)^k = 0 when ω^j ≠ 1: multiplying the sum S by ω^j gives ω^jS = S, forcing S = 0.
• Number of primitive roots: φ(n), Euler's totient.
• For prime p, the minimal polynomial of a primitive root is 1 + x + … + x^{p−1}.

Worked Example

Find the cube roots of unity: 1, ω = cos(2π/3) + i sin(2π/3) = −1/2 + i√3/2, and ω² = −1/2 − i√3/2. Verify ω³ = 1: ω² = (−1/2 + i√3/2)² = 1/4 − 3/4 − i√3/2 = −1/2 − i√3/2, then ω³ = ω²·ω = (−1/2 − i√3/2)(−1/2 + i√3/2) = 1/4 + 3/4 = 1. Verify the sum: 1 + ω + ω² = 1 + (−1/2 + i√3/2) + (−1/2 − i√3/2) = 0. The equilateral-triangle symmetry forces the vanishing sum.

Common Misconceptions

• "The n-th root of 1 is 1." That is just one of n roots; the other n − 1 sit on the unit circle.
• "Every root of unity is primitive." Only exponents coprime to n qualify; e^{2πi·2/6} has order 3, not 6.
• "The sum vanishes by pairwise cancellation." That intuition fails for odd n; the real reason is rotational symmetry: multiplying the sum by ω leaves it unchanged, so it must be zero.

Real-World Applications

• The FFT evaluates polynomials at roots of unity, driving audio/image compression and fast large-integer multiplication.
• Antenna arrays and digital filters exploit root-of-unity symmetry to shape frequency responses.
• Cyclotomic fields built from roots of unity were central to work on Fermat's Last Theorem.

Roots of unity compress perfect symmetry into n points on the unit circle, and their vanishing sum plus the filtering identity turn that symmetry into a computational workhorse.