Product of nth Roots of Unity
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Theorem
Let $n \in \Z$ be an integer such that $n > 0$.
Let $z \in \C$ be a complex number such that $z^n = 1$.
Then:
- $U_n = \set {e^{2 i k \pi / n}: k \in \N_n}$
where $U_n$ is the set of $n$th roots of unity.
That is:
- $z \in \set {1, e^{2 i \pi / n}, e^{4 i \pi / n}, \ldots, e^{2 \paren {n - 1} i \pi / n} }$
Then the product of all of the elements of $U_n$ is $(-1)^{n-1}$
Proof
\(\ds \prod_{k \mathop = 0}^{n - 1} e^{2 i k \pi / n}\) | \(=\) | \(\ds e^{2 i \paren {0} \pi / n} e^{2 i \paren {1} \pi / n} e^{2 i \paren {2} \pi / n} \dotsm e^{2 i \paren {n - 1} \pi / n}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds e^{\paren {2 i \pi / n} \paren {0 + 1 + \dotsm + \paren {n - 1} } }\) | Product of Powers | |||||||||||
\(\ds \) | \(=\) | \(\ds e^{\paren {2 i \pi / n} \paren {\frac {n \paren {n - 1} } 2 } }\) | Closed Form for Triangular Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds e^{\paren {i \pi } \paren {n - 1 } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-1}^{\paren {n - 1 } }\) | $e^{i \pi} = -1$ |
$\blacksquare$