Note that Maple uses the uppercase letter I, rather than the lowercase letter i, to denote the imaginary unit: .
Since is a polynomial with complex coefficients and a degree of n, it must have exactly n complex roots according to the Fundamental Theorem of Algebra.
To solve for all the roots of unity, we will use de Moivre's Theorem: , where x is any complex number and n is any integer (in this particular case x will be any real number and n will be any positive integer).
First, convert the complex number z to its polar form: , where is the modulus of z and θ is the angle between the positive real axis (Re) and the line segment joining the point z to the origin on the complex plane. Since , it must be true that , and so the previous equation simply becomes .
Also, converting the real number to polar form, we get for any integer k.
Now, and so using de Moivre's Theorem, this equation becomes . From this form of the equation, we can see that , or equivalently, .
Therefore, the roots of unity can be expressed using the formula , for .
Using Euler's formula: , we can write this formula for the roots of unity in its most common form: , for .