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This essential trains for: AMC-12, AIME.

This is an equation in which the solutions are n-th roots of a complex number.

Given a complex number z, the equation:

equation

is cyclotomic.

In order to solve this equation we have to write 1 in trigonometric (or exponential) form because in any of these two forms we can easily take the n-th root using the de Moivre formula.

The absolute value of 1 is 1 and the argument 0. Taking into account the periodicity of the trigonometric functions:

equation

apply de Moivre:

equation

Notice that:

equation

is 1, the one real root of the equation.

The next root:

equation

can be represented as a point on the unit circle that is one n-th of a complete rotation in counter-clockwise direction.

Each subsequent root will be rotated by the same arc and the n-th root will finally coincide with the first one:

equation

which means that there are only n distinct values of the argument of the generic root. Therefore, there are only n distinct roots.

These roots all have the same absolute value, which means they are all points on a unit circle in the complex plane.

The roots are placed on the circle at constant intervals of:

equation

effectively "cutting the circle" into n congruent sectors. Hence, the name "cyclotomic."

Example 1: Find the roots of the equation:

equation

equation

figure

The roots are the vertices of a regular pentagon inscribed in the unit circle.

There is one real root and 2 pairs of complex conjugate roots.

Properties of the roots

According to Viète's relations, the sum of the roots must be zero:

equation

which is, in fact, the same as:

equation

since the imaginary parts are pairs of opposite numbers and add up to zero.

The product of all the roots must be equal to:

equation

The sum of the squares of the lengths of all the diagonals of the regular n-gon formed, is 2n. Proof:

The square of the length of the diagonal from a k-th root to the 0-th root, which is 1+0i is:

equation

equation

equation

equation

add over all diagonals:

equation

equation

and use Viète's relation for the sum of the roots:

equation

Obviously, since:

equation

the non-real roots of the cyclotomic equation are the roots of:

equation