EXPLANATION OF CUBE ROOT OF UNITY !!

The cube roots of unity are the solutions to the equation

  z³ = 1

in the complex numbers. In other words, they are the numbers z that, when raised to the third power, equal 1. These roots have rich algebraic and geometric significance, appearing in areas ranging from number theory to signal processing.

Derivation Using Euler’s Formula

Euler’s formula states that for any real number θ,
  e^(iθ) = cos θ + i sin θ

Since we want z³ = 1, we can write 1 in its exponential form as e^(2πi·m) for any integer m (because e^(2πi) = 1). Therefore, we set

  z³ = e^(2πi·m)

Taking the cube root of both sides gives

  z = e^(2πi·m/3)

Since complex exponentials are periodic with period 2π, we only need to consider three distinct values corresponding to m = 0, 1, and 2.

The Three Cube Roots of Unity

1. For m = 0:

  z = e^(2πi·0/3) = e^0 = 1

2. For m = 1:

  z = e^(2πi/3) = cos(2π/3) + i sin(2π/3)
      = -1/2 + i (√3/2)

3. For m = 2:

  z = e^(4πi/3) = cos(4π/3) + i sin(4π/3)
      = -1/2 - i (√3/2)

Thus, the three cube roots of unity are:

• ω₀ = 1
• ω₁ = (-1 + √3 i)⁄2
• ω₂ = (-1 - √3 i)⁄2

Geometric Interpretation

On the complex plane, these three points lie on the unit circle (since |e^(iθ)| = 1 for any θ). They are equally spaced at angles of 0, 120, and 240 degrees (or equivalently, 0, 2π/3, and 4π/3 radians). This equal spacing means that if you plot the cube roots of unity, they form the vertices of an equilateral triangle inscribed in the unit circle.

Key Properties

1. Symmetry and Conjugation:
     ω₁ and ω₂ are complex conjugates of each other. This is because the sine function is odd and the cosine function is even, so the imaginary parts of e^(2πi/3) and e^(4πi/3) are negatives of one another while their real parts are the same.

2. Sum of the Roots:
     One important identity is:
       1 + ω₁ + ω₂ = 0
     This result can be verified directly by adding the three roots. It also follows from Vieta’s formulas applied to the polynomial x³ - 1 = 0 (after factoring out x - 1).

3. Multiplicative Structure:
     The cube roots of unity form a cyclic group under multiplication. Multiplying any two of these roots gives another cube root of unity. For example,
       ω₁ · ω₁ = ω₂
     and
       ω₁ · ω₂ = 1.

4. Minimal Polynomial:
     For the non-real cube roots, the minimal polynomial over the reals is x² + x + 1 = 0. This is because if z is a primitive cube root of unity (i.e., not equal to 1), then z satisfies:
       z² + z + 1 = 0

Applications

The cube roots of unity are more than just an abstract mathematical curiosity—they have practical applications:

• Algebra and Number Theory:
  The cube roots of unity are used in the factorization of polynomials. For example, x³ - 1 can be factored as (x - 1)(x² + x + 1), where x² + x + 1 has the cube roots of unity (other than 1) as its solutions.

• Signal Processing:
  In the context of the discrete Fourier transform (DFT), roots of unity play a crucial role in transforming signals from the time domain to the frequency domain. Their properties allow for efficient algorithms like the Fast Fourier Transform (FFT).

• Geometry and Symmetry:
  The geometric arrangement of the roots of unity exemplifies symmetry in the complex plane. This symmetry underlies many constructions in both pure and applied mathematics.

Further Insights

When generalizing from cube roots of unity to nth roots of unity, the pattern remains similar. The nth roots of unity are given by:

  z = e^(2πik/n) for k = 0, 1, 2, …, n – 1

These roots are equally spaced on the unit circle and form the vertices of a regular n-sided polygon. The study of these roots leads into the rich field of cyclotomic polynomials, which have deep implications in number theory and field theory.

The concept of "primitive" roots is also essential here. A primitive nth root of unity is one that is not also an mth root of unity for some m < n. For the cube roots of unity, both ω₁ and ω₂ are primitive because their powers generate all the distinct cube roots (1, ω₁, ω₂). In contrast, 1 is not considered primitive since its powers do not generate new values.

Conclusion

In summary, the cube roots of unity are the three complex numbers—1, (-1 + √3 i)/2, and (-1 - √3 i)/2—that satisfy z³ = 1. They are elegantly derived using Euler’s formula, have a beautiful geometric representation as the vertices of an equilateral triangle on the unit circle, and possess properties that are fundamental in various areas of mathematics. The identity 1 + ω₁ + ω₂ = 0 is a key insight that not only simplifies algebraic manipulations but also underscores the inherent symmetry of these numbers. Understanding these roots provides a gateway to more advanced topics such as cyclotomic polynomials and the discrete Fourier transform, making them a central concept in both theoretical and applied mathematics.