Root of Unity - Examples

Examples

De Moivre's formula, which is valid for all real x and integers n, is

Setting x = 2π/n gives a primitive nth root of unity:

but for k = 1, 2, ... n−1,

This formula shows that on the complex plane the nth roots of unity are at the vertices of a regular n-sided polygon inscribed in the unit circle, with one vertex at 1. (See the plots for n = 3 and n = 5 on the right). This geometric fact accounts for the term "cyclotomic" in such phrases as cyclotomic field and cyclotomic polynomial; it is from the Greek roots "cyclo" (circle) plus "tomos" (cut, divide).

Euler's formula

which is valid for all real x, can be used to put the formula for the nth roots of unity into its most familiar form

It follows from the discussion in the previous section that this is a primitive root if and only if the fraction k/n is in lowest terms, i.e. that k and n are coprime.

The roots of unity are, by definition, the roots of a polynomial equation and are thus algebraic numbers. In fact, Galois theory can be used to show that they may be expressed as expressions involving integers and the operations of addition, subtraction, multiplication, division, and the extraction of roots. (There are more details later in this article at Cyclotomic fields.)

The equation z1 = 1 obviously has only one solution, +1, which is therefore the only primitive first root of unity. It is a nonprimitive 2nd, 3rd, 4th, ... root of unity.

The equation z2 = 1 has two solutions, +1 and −1. +1 is the primitive first root of unity, leaving −1 as the only primitive second (square) root of unity. It is a nonprimitive 4th, 6th, 8th, ...root of unity.

The only real roots of unity are ±1; all the others are non-real complex numbers, as can be seen from de Moivre's formula or the figures.

The third (cube) roots satisfy the equation z3 − 1 = 0; the non-principal root +1 may be factored out, giving (z − 1)(z2 + z + 1) = 0. Therefore, the primitive cube roots of unity are the roots of a quadratic equation. (See Cyclotomic polynomial, below.)

The two primitive fourth roots of unity are the two square roots of the primitive square root of unity, −1

The four primitive fifth roots of unity are

The two primitive sixth roots of unity are the negatives (and also the square roots) of the two primitive cube roots:

Gauss observed that if a primitive nth root of unity can be expressed using only square roots, then it is possible to construct the regular n-gon using only ruler and compass, and that if the root of unity requires third or fourth or higher radicals the regular polygon cannot be constructed. The 7th roots of unity are the first that require cube roots. Note that the real part and imaginary part are both real numbers, but complex numbers are buried in the expressions. They cannot be removed. See casus irreducibilis for details.

One of the primitive seventh roots of unity is

where ω and ω2 are the primitive cube roots of unity exp(2πi/3) and exp(4πi/3).

The four primitive eighth roots of unity are ± the square roots of the primitive fourth roots, ±i. One of them is:

See heptadecagon for the real part of a 17th root of unity.