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This essential trains for: AMC-10, AMC-12, AIME.

The cubic roots of unity are the solutions of the polynomial equation:

equation

This is a cyclotomic equation - see the lesson on cyclotomic equations.

We expect it to have three complex roots, since its degree is 3.

To solve this equation, write the number 1 in trigonometric form:

equation

Take the third root by applying the de Moivre formula:

equation

Since the expression of the roots is periodic, there are three distinct values for the roots, given by k=0, 1, 2:

equation

equation

equation

However, it is easy to see that the third root is the complex conjugate of the second one, due to elementary trigonometric transformations:

equation

equation

equation

Traditionally, the complex-imaginary roots of unity have been denoted with ω :

equation

equation

Therefore, the roots of one are:

equation

Since, according to De Moivre's formula:

equation

we can also write them as:

equation

Which is pretty obvious when we think of the algebraic identity:

equation

which shows by factoring that the one real root is equal to 1 and the two complex-imaginary roots are roots of the quadratic equation:

equation

Properties of the complex roots of unity

From the above discussion:

equation

equation

equation

equation

Problem (similar to a problem from a 1959 Moscow Olympiad), illustrating the connection to polynomials:

If the polynomial

equation

is expanded into a canonical form:

equation

what is the value of the expression:

equation

Solution We notice that:

equation

equation

equation

equation

equation

and so on. The real parts of these powers of ω form the sequence:

equation

and the imaginary parts form the sequence:

equation

The sequence of real parts corresponds exactly to the coefficients of the expression we have to compute. We realize that the expression can be computed by calculating the value of the polynomial at ω and taking the real part:

equation

To compute, we use the properties of the cubic roots of unity:

equation

equation

equation

equation

and the that could be written for the imaginary part is zero for this particular polynomial:

equation