Roots of Unity

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:

$U_n = \left\{{e^{2 i k \pi / n}: k \in \N_n}\right\}$

where $U_n$ is the set of $n$th roots of unity.

That is:

$z \in \left\{{1, e^{2 i \pi / n}, e^{4 i \pi / n}, \ldots, e^{2 \left({n-1}\right) i \pi / n}}\right\}$

Thus for every integer $n$, the number of $n$th roots of unity is $n$.

First Root of Unity

The root $e^{2 i \pi / n}$ is known as the first $n$th root of unity.

Proof

Let $z \in \left\{{e^{2 i k \pi / n}: k \in \N_n}\right\}$.

Then $z^n \in \left\{{e^{2 i k \pi}: k \in \N_n}\right\}$.

Hence $z^n = 1$.

Now suppose $z^n = 1$.

Let $z = r e^{i \theta}$.

Then $\left|{z^n}\right| = 1 \implies \left|{z}\right| = 1$.

Similarly, we have $n \theta = 0 \bmod 2 \pi$.

So $\theta = \dfrac {2 k \pi} n$ for $k \in \Z$.

Hence the result.

$\blacksquare$

Sources

• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): Introduction