Root of unity

In algebra, numbers whose -th power be in the number 1, called te roots of unity.

• 3.1 unit roots in bodies
• 3.2 unit roots in residue class rings

Definition

It is a commutative ring with unit element and a natural number. An element is called a root of unity if it meets one of the two equivalent conditions:

• ;
• Is zero of the polynomial.

The roots of unity in a subgroup of the multiplicative group, which is often referred to.

A root of unity is called primitive if for applies.

Roots of unity in the complex numbers

Are in the field of complex numbers

The roots of unity. Substituting

So is primitive, and these numbers get ( in the same order ) the simple form

Is it clear to which it is, you can often drop the subscript. The nth root is then exactly primitive if and are relatively prime.

Group of roots of unity

As with and also roots of unity, the set of all roots of unity is a group. The illustration:

Is surjective. The core of this picture is. The group of complex roots of unity is therefore isomorphic to the factor group.

Geometric relation

The roots of unity can be used in the complex plane geometric interpretation clearly: You are on the unit circle lying ( with center 0 and radius 1) vertices of a regular -gon, one of the corners is the number because this is for each one - th root of unity.

Real and imaginary part of the roots of unity so that the coordinates of the corners of the pentagon in the circuit, ie is for

More see root extraction of complex numbers.

Sum of roots of unity

Is a root of unity, it shall

This statement follows immediately from the geometric sum formula and is a special case of the analogous statement for characters of groups.

Examples

The second, third and fourth roots of unity

The second unit roots are

The third roots of unity are

The fourth roots of unity are back from a simpler form:

Where i is the imaginary unit.

The fifth roots of unity

It follows from

For. Provides Solve this quadratic equation. Because the angle in the first quadrant, is positive, and thus the real part of. The imaginary part is according to the Pythagorean theorem.

Properties of the roots of unity

Roots of unity in bodies

In a body the roots of unity form a cyclic subgroup of the multiplicative group. Their number is always a divisor of. If it is equal, it is said, " contains the roots of unity ."

Contains the roots of unity, as a root of unity if and only primitive if it generates the group of roots of unity. The primitive roots of unity are precisely the zeros of th Kreisteilungspolynoms.

Enhancements arising from adjunction of roots of unity, called cyclotomic fields.

Roots of unity in residue class rings

• In the ring of integers modulo the number is a primitive root of unity, as is true in this ring.
• In the ring of integers modulo the number is a primitive root of unity.

These two specific residue class rings are highly significant for computer algebra, because they allow an even more drastically accelerated version of the fast discrete Fourier transform. This is due to the fact that the addition and multiplication of the residual rings can be replaced by corresponding cyclic addition and multiplication in a slightly larger residue class ring, and therefore in binary number representation of the multiplication by powers of the number represents a cyclic binary shift operation, which is much quicker to carry out as a general multiplication of two numbers. The significant time savings for the discrete Fourier transform arises from the fact that many multiplications are carried out with the selected unit root during the fast Fourier transform.