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Primitive Root of Unity — Definition, Formula & Examples

A primitive nnth root of unity is a complex number ω\omega such that ωn=1\omega^n = 1 but ωk≠1\omega^k \neq 1 for any positive integer k<nk < n. In other words, it is a root of unity whose order is exactly nn, meaning you must raise it to the full nnth power before it first equals 1.

A complex number ω\omega is a primitive nnth root of unity if ωn=1\omega^n = 1 and ord⁡(ω)=n\operatorname{ord}(\omega) = n, i.e., nn is the smallest positive integer for which ωn=1\omega^n = 1. Equivalently, ω=e2πik/n\omega = e^{2\pi i k/n} where gcd⁡(k,n)=1\gcd(k, n) = 1 and 1≤k≤n−11 \le k \le n - 1. The powers ω0,ω1,…,ωn−1\omega^0, \omega^1, \dots, \omega^{n-1} are all distinct and form the complete set of nnth roots of unity.

Key Formula

ω=e2πik/n,gcd⁡(k,n)=1\omega = e^{2\pi i k / n}, \quad \gcd(k, n) = 1
Where:
  • ω\omega = A primitive nth root of unity
  • nn = A positive integer specifying which roots of unity are considered
  • kk = An integer between 1 and n−1 that is coprime to n

How It Works

The standard nnth root of unity is ω=e2πi/n\omega = e^{2\pi i / n}, which is always primitive. To find all primitive nnth roots, take e2πik/ne^{2\pi i k/n} for each kk with gcd⁡(k,n)=1\gcd(k,n) = 1. The number of primitive nnth roots of unity equals φ(n)\varphi(n), Euler's totient function. A non-primitive root of unity has order dividing nn but strictly less than nn, so its powers cycle back to 1 before generating all nn roots.

Worked Example

Problem: Find all primitive 6th roots of unity.
Step 1: Identify which values of k from 1 to 5 satisfy gcd(k, 6) = 1.
gcd⁡(1,6)=1,  gcd⁡(5,6)=1\gcd(1,6)=1,\; \gcd(5,6)=1
Step 2: The other values k = 2, 3, 4 share a common factor with 6, so they give non-primitive roots.
gcd⁡(2,6)=2,  gcd⁡(3,6)=3,  gcd⁡(4,6)=2\gcd(2,6)=2,\; \gcd(3,6)=3,\; \gcd(4,6)=2
Step 3: Write out the two primitive 6th roots of unity using the formula.
ω1=e2πi/6=eiπ/3=12+32 i,ω5=e10πi/6=e5iπ/3=12−32 i\omega_1 = e^{2\pi i/6} = e^{i\pi/3} = \frac{1}{2} + \frac{\sqrt{3}}{2}\,i, \quad \omega_5 = e^{10\pi i/6} = e^{5i\pi/3} = \frac{1}{2} - \frac{\sqrt{3}}{2}\,i
Answer: The primitive 6th roots of unity are eiπ/3=12+32 ie^{i\pi/3} = \tfrac{1}{2} + \tfrac{\sqrt{3}}{2}\,i and e5iπ/3=12−32 ie^{5i\pi/3} = \tfrac{1}{2} - \tfrac{\sqrt{3}}{2}\,i. There are φ(6)=2\varphi(6) = 2 of them.

Why It Matters

Primitive roots of unity are central to the discrete Fourier transform (DFT) used in signal processing and the FFT algorithm. In number theory, they connect to cyclotomic polynomials, which factor xn−1x^n - 1 over the rationals and appear in proofs about constructibility of regular polygons.

Common Mistakes

Mistake: Assuming every nth root of unity is primitive.
Correction: Only roots e2πik/ne^{2\pi i k/n} with gcd⁡(k,n)=1\gcd(k,n) = 1 are primitive. For example, ii is a 4th root of unity and primitive, but −1-1 is also a 4th root of unity yet not primitive since (−1)2=1(-1)^2 = 1 (its order is 2, not 4).