Properties of the Cube Root of Unity

The complex cube roots of unity have three exact roots, 1, −1+i√3/2, −1−i√3/2, which are composed of 1, ω, ω2 . 

To put it simply, the complex root of unity has one real root, which is 1, and two imaginary roots, which are and.

The imaginary root of unity is symbolised as omega, and hence the other as square omega. The product or multiple of the three cube roots of unity is generally 1 (1.ω.ω2  = ω3 = 1). Therefore, the total of the cube roots of unity is frequently close to zero (1 + ω + ω2  = 0).

Properties of the complex cube root of unity

The following are a number of the important properties of the root of unity.

• The complex cube root of unity has omega and omega square as the two imaginary roots (ω, ω2 ) and one of the real roots, which is 1.
• The sum of the imaginary roots that is omega and omega square and the real root that is one is equal to zero.(1 + ω + ω2  = 0)
• The square of 1 imaginary root omega (ω) of the root of unity is equal to another imaginary root omega square (ω2 ) of the root of unity.
• The product of the imaginary roots  that is omega and omega square of the complex cube roots of unity is equal to 1. (ω.ω2  = ω3 = 1)
• If we look at the three complex cube roots of unity, we can say that among the three cube roots, two of them are the imaginary roots symbolism as omega and omega square and the other one is a real cube root.
• The three complex cube roots of unity are stated as 1, -1/2 + i√3/2 and -1/2 – i√3/2.
• We can establish that the complex cube root of unity we get is 1, which is the real cube root and the other two, i.e., 1/2 + i√32 and -1/2 – i√32, which are the imaginary complex cube root numbers.
• The square of the omega imaginary complex cube root of unity is equal to the other omega imaginary root of unity.

Hence, we conclude that the square of any root of unity is equal to the other.

If ω2  is one imaginary root of unity, then the other would be ω.

• The product of the two imaginary cube roots is 1, or the total product of three cube roots of unity is 1.
• The cube roots of unity are 1, ω, ω2 . 
• Thus, the product of cube roots of unity is 1.

 ∙ ω ∙ ω2  = ω3 = 1. 

• Therefore, the product of the three cube roots of unity is 1.

ω3 = 1

• Since ω is a root of the equation z3 – 1 = 0, ω satisfies the equation z3 – 1 = 0. 

Consequently, ω3 – 1 = 0

or, ω = 1.

Note: Since ω3 = 1, ωn = ωm, where m is the least non-negative remainder obtained by dividing n by 3.

• The sum of the three cube roots of unity is zero i.e., 1 + ω + ω2  = 0.

In every subject, the roots of unity are frequently defined. If the sector’s characteristic is zero, the roots are complex numbers that are also algebraic integers. The roots of fields with a positive characteristic belong to a finite field and vice versa. Any nonzero element of a finite field can be a root of unity. Except when a field is algebraically closed, it includes precisely n nth roots of unity.

Important properties of the cube root of unity

• The two complex cube roots of the unity are imaginary.
• The complex cube root is the complex root to another one.
• If we state the cube root of unity and denote the root as w to be one complex root, then the three complex roots are 1, w, ω2 .
• Since w may be a root of equation x^3=1, we obtain w^3=1, where w 4=w, w5=ω2 , w^6=(w^3)^2=1, i.e., w^{3m}=1, w^{3m+1}=w, w^{3m+2}=ω2 , m \in N.
• 1+w+ω2 =0, i.e., the sum of the three cube roots of unity vanishes.

Conclusion

The imaginary root of unity is represented by a logo known as omega, and thus the other one as square omega. The multiple or the product of the three complex cube roots of unity is typically 1 (1.ω.ω2  = ω3 = 1). The sum of the cube roots of unity is typically zero. (1 + ω + ω2  = 0).