Overview

Advanced root-of-unity problems reduce to divisibility in the exponents or cyclotomic factorizations.

Core Skills

Use the Power Sum Filter

For $\omega$ an $n$th root of unity, $\sum_{k=0}^{n-1}\omega^{km}$ is $n$ if $n\mid m$ and $0$ otherwise.

Separate Primitive Roots

Group roots by their order to control which terms survive in a sum.

Check Reality Conditions

Pair conjugate roots to determine when a sum is real.

Key Ideas

• Power sum: $\sum_{k=0}^{n-1} \omega^{km} = n$ if $n\mid m$, otherwise $0$.
• Primitive roots organize factors of $x^n-1$.
• Reality conditions become angle divisibility checks.

Worked Example

Let $\omega = e^{2\pi i/6}$. Compute $\sum_{k=0}^{5} \omega^{3k}$.

Since $\omega^3=-1$, the sum is $1-1+1-1+1-1=0$.

More Examples

Example 1: Divisibility Filter

Let $\omega$ be a primitive $7$th root of unity. Find $\sum_{k=0}^{6}\omega^{2k}$.

Since $7\nmid 2$, the sum is $0$.

Example 2: Real Sum

Compute $\omega+\omega^{-1}$ for $\omega=e^{2\pi i/5}$.

It equals $2\cos(2\pi/5)$.

Example 3: Cyclotomic Factor

Factor $x^6-1$ over the reals.

$(x-1)(x+1)(x^2+x+1)(x^2-x+1)$.

Strategy Checklist

• Check whether the exponent is divisible by $n$.
• Pair conjugates to keep sums real.
• Use primitive roots to simplify factors.

Common Pitfalls

• Forgetting to test whether $n$ divides the exponent.
• Dropping conjugate pairs when checking real-valued sums.
• Mixing primitive and non-primitive roots in sums.

Practice Problems