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De-Moivre's Theorem

If 1, α1, α...

Question

If $1, α_{1}, α_{2}, α_{3}, . . . . ., α_{n - 1}$ are the nth roots of unity then show that $n = (1 - α_{1}) (1 - α_{2}) (1 - α_{3}) . . . . . (1 - α_{n - 1})$ .

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Solution

Given n, nth roots of unity are
$1, α_{1}, α_{2}, α_{3}, . . . . ., α_{n - 1}$
Let $x = (1)^{1 / n}$
$\Rightarrow x^{n} - 1 = 0$
$\Rightarrow x^{n} - 1 = (x - 1) (x - α_{1}) (x - α_{2}) (x - α_{3}) . . . . . (x - α_{n - 1})$
$\Rightarrow \frac{x^{n} - 1}{x - 1} = (x - α_{1}) (x - α_{2}) (x - α_{3}) . . . . (x - α_{n - 1})$
Taking $lim x \to 1$ on both sides, we have
$lim x \to 1 \frac{x^{n} - 1}{x - 1} = lim x \to 1 (x - α_{1}) (x - α_{2}) (x - α_{3}) . . . . . (x - α_{n - 1})$
$\Rightarrow n = (1 - α_{1}) (1 - α_{2}) (1 - α_{3}) . . . . (1 - α_{n - 1})$

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