Product of Differences between 1 and Complex Roots of Unity
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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 $5$.