Overview

Roots of unity are equally spaced points on the unit circle. Their symmetry collapses many sums to $0$.

Core Skills

Parameterize the Roots

Write $z_k=e^{2\pi i k/n}$ and use symmetry of the regular $n$-gon.

Use the Geometric Sum

Apply $1+z+\cdots+z^{n-1}=0$ when $z^n=1$ and $z\ne 1$.

Filter by Exponents

If a sum includes $z^{mk}$, reduce $m$ modulo $n$ to see which terms survive.

Key Ideas

• $z^n=1$ gives $z_k = e^{2\pi i k/n}$.
• The roots form a regular $n$-gon centered at the origin.
• $1 + z + z^2 + \cdots + z^{n-1} = 0$ for $z^n=1$, $z\ne 1$.

Worked Example

If $z^5=1$ and $z\ne 1$, compute $1+z+z^2+z^3+z^4$.

This is the full sum of the 5th roots except $1$, so it equals $0$.

More Examples

Example 1: Sum of All Roots

Find the sum of all cube roots of unity.

It is $0$.

Example 2: Power Sum

If $\omega^7 = 1$ and $\omega \ne 1$, find $1 + \omega^2 + \omega^4 + \omega^6$.

This is half the roots, so the sum is $-1$.

Example 3: Geometry Interpretation

What is the sum of the vertices of a regular $n$-gon centered at the origin?

By symmetry, it is $0$.

Example 4: Roots of Unity Trick (remember this one!)

Let $\omega \ne 1$ be a 13th root of unity. Your job is to compute the following expression: $\prod_{k=1}^{12}(1 - \omega^k)$.

Using the following trick: $x^{13} - 1 = (x - 1)(x^{12} + x^{11} + \cdots + 1)$ and substituting in $x = 1$, we get that the product is $13$.

Example 5: Roots of Unity Reciprocals (also very common, so be sure to understand this completely and be able to recognize this!)

Given that if $\omega^7 = 1$ and also $\omega \ne 1$, find the following: $\sum_{k=0}^{6} \frac{1}{1 + \omega^k}$.

The trick here is to pair terms $\frac{1}{1 + \omega^k}$ and $\frac{1}{1 + \omega^{-k}}$ gives $1$ for each pair, and the $k = 0$ term is $\frac{1}{2}$. So, we get that the total is simply $\frac{7}{2}$.

Example 6: Roots of Unity + Recursion (think out of the box!)

Let $\omega$ be a primitive cube root of unity. Your job is to find $(2 + \omega)^{10} + (2 + \omega^2)^{10}$.

First of all, since $2 + \omega$ and $2 + \omega^2$ are roots of $x^2 - 3x + 3 = 0$, the we can see that the sequence satisfies a linear recurrence, and working off this gives $243$.

Strategy Checklist

• Write roots in exponential form.
• Use geometric series for full sums.
• Reduce exponents modulo $n$.

Common Pitfalls

• Forgetting to exclude the root $z=1$.
• Treating the sum as $5$ instead of $0$.
• Forgetting to reduce exponents modulo $n$.

Practice Problems