Sum and Product of nth Roots of Unity

The sum of all nth roots of unity is always zero — geometrically, the n equally spaced vectors cancel. The product is (−1)^(n+1): it equals +1 for odd n and −1 for even n. Both follow from Vieta's formulas applied to zⁿ − 1 = 0.

The $n$th roots of unity are the $n$ solutions of $z^n = 1$, equivalently $z^n - 1 = 0$. Their sum and product are determined entirely by the coefficients of this polynomial — no computation needed.

Vieta's Formulas Applied

The polynomial $z^n - 1 = 0$ has roots $1, \omega, \omega^2, \ldots, \omega^{n-1}$ where $\omega = e^{2\pi i/n}$.

Written in factored form:

$$z^n - 1 = (z - 1)(z - \omega)(z - \omega^2)\cdots(z - \omega^{n-1})$$

Expanding the right side, the coefficient of $z^{n-1}$ is $-(1 + \omega + \omega^2 + \cdots + \omega^{n-1})$, and the constant term is $(-1)^n \cdot 1 \cdot \omega \cdot \omega^2 \cdots \omega^{n-1}$.

Comparing with $z^n + 0 \cdot z^{n-1} + \cdots + 0 \cdot z - 1$:

Sum (coefficient of $z^{n-1}$ is 0):

$$1 + \omega + \omega^2 + \cdots + \omega^{n-1} = 0$$

Product (constant term is $-1$):

$$1 \cdot \omega \cdot \omega^2 \cdots \omega^{n-1} = (-1)^{n+1}$$

The product equals $+1$ for odd $n$ and $-1$ for even $n$.

Geometric Confirmation of the Sum

The $n$ roots are equally spaced on the unit circle at angles $0, 2\pi/n, 4\pi/n, \ldots$. Their sum as vectors:

$$\sum_{k=0}^{n-1} e^{2\pi ik/n}$$

By symmetry, this vector sum must point in no particular direction — the only such vector is $\mathbf{0}$. The algebraic result and the geometric intuition agree.

The Product via Exponents

$$\prod_{k=0}^{n-1} \omega^k = \omega^{0+1+2+\cdots+(n-1)} = \omega^{n(n-1)/2} = e^{2\pi i \cdot n(n-1)/(2n)} = e^{i\pi(n-1)}$$ $$= \cos\pi(n-1) + i\sin\pi(n-1) = (-1)^{n-1} = (-1)^{n+1}$$

Consistent with Vieta's result. $\blacksquare$

Partial Sums

For any $m$ not divisible by $n$:

$$\sum_{k=0}^{n-1} \omega^{mk} = \frac{\omega^{mn} - 1}{\omega^m - 1} = \frac{1 - 1}{\omega^m - 1} = 0$$

This generalisation — the sum of any $n$ equally-spaced points on the unit circle is zero — is used in problems involving selective sums like $1 + \omega^2 + \omega^4 + \cdots$.