n-th roots

Definition

Definition

• If $z \in \mathbb C$, we call n-th root of $z$ any $Z \in \mathbb C$ such that $Z^n = z$.
• The n-th roots of one are also called root of unity.

Root of unity

Proposition

It exists exactly $n$ root n-th of unity, which are the complexes

$$\xi_k = e^{i\frac{2k\pi}{n}} = \xi_1^k, \quad \text{avec } k \in [0, n-1] \subset \mathbb N.$$

Proposition

Let $n$ be an integer greater than or equal to 2.

• if $\xi$ is a n-th root of unity not equal to 1, we have

$$1 + \xi_1 + \xi_2 + \dots + \xi^{n-1} = 0.$$

• The sum of n-th roots of unity is equal to 0.

n-th root

Proposition

If $Z_0$ si a n-th root of $z$, the set of n-th root of $z$ is

$$\{ Z_0\xi; \xi \in \mathbb U_n \}.$$

Proposition

Let $n \neq 0$ be an integer and let $z \neq 0$ be complex number of module $r$ and with argument $\theta$, the complex number $z$ admits $n$ n-th roots which are the complex numbers :

$$Z_k = r^{\frac{1}{n}}e^{i(\frac{\theta}{n} + \frac{2k\pi}{n})} \quad \text{ with } \quad k \in [0, n-1] \subset \mathbb N.$$