Positive Real Complex Root of Unity

Theorem

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

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


The only $x \in U_n$ such that $x \in \R_{>0}$ is:

$x = 1$


That is, $1$ is the only complex $n$th root of unity which is a positive real number.


Proof

We have that $1$ is a positive real number.

The result follows from Existence and Uniqueness of Positive Root of Positive Real Number.

$\blacksquare$


Sources

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