Product of Differences between 1 and Complex Roots of Unity

Theorem

Let $\alpha$ be a primitive complex $n$th root of unity.

Then:

$\ds \prod_{k \mathop = 1}^{n - 1} \paren {1 - \alpha^k} = n$

Proof

From Power of Complex Number minus 1: Corollary:

$\ds \sum_{k \mathop = 0}^{n - 1} z^k = \prod_{k \mathop = 1}^{n - 1} \paren {z - \alpha^k}$

The result follows by setting $z = 1$.

$\blacksquare$

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 3$. Roots of Unity: Example $5$.