##

Roots of unity.

**Definition 1.5.4**
Let

be a natural number such that

. A complex number

such that

is called an

root of unity.

By the method of subsection 5.1, we compute all the
roots of unity:

For
, we have the following roots:

The images in Gauss-Argand plane of the
roots of unity are the vertices of a regular polygone inscribed in the unit circle.

By De Moivre's formula De Moivre's formula, the following holds:

An immediate consequence is as follows:

**Proposition 1.5.7**
*Let
be a natural number such that
. The sum of all
roots of unity is equal to 0.*

*Proof*.
We have:

Now, let us see a nice application of these roots of unity.

**Proposition 1.5.8**
*Let
be a natural number such that
. Suppose that
and
are two complex numbers such that
. Then we obtain all the
roots of
by separate multiplication of
by all the
roots of unity.*

**Example 1.5.9**
Take the number

Squarring twice, we obtain

By Proposition

5.8, the

roots of

are given by:

An additional result of this example is the following: compute polar forms for the
roots of
. Theses roots have arguments
,
,
and
. They fit exactly the order of the roots given above.

Noah Dana-Picard
2007-12-24