Sum of Products of nth Roots of Unity taken up to n-1 at a Time is Zero

Theorem

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $U_n = \set {z \in \C: z^n = 1}$ be the complex $n$th roots of unity.

Then the sum of the products of the elements of $U_n$ taken $2, 3, 4, \dotsc n - 1$ at a time is zero.

Proof

The elements of $U_n = \set {z \in \C: z^n = 1}$ are the solutions to the equation:

$z^n - 1 = 0$

Thus by definition the coefficients of the powers of $z$:

$z^2, z^3, \ldots, z^{n - 1}$

are all zero.

The result follows directly from Viète's Formulas.

$\blacksquare$

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: The $n$th Roots of Unity: $108$