Let \(\omega\) be a root of \(x^2 - x + 1 = 0\). Then \(\omega^2 = \omega - 1\), and \(\omega^3 = \omega(\omega - 1) = \omega^2 - \omega = (\omega - 1) - \omega = -1\). So \(\omega^3 = -1\), and \(\omega^6 = 1\).

["Understanding (\omega): A Root of the Equation (x^2 - x + 1 = 0)", "In the realm of complex numbers and algebraic structures, certain roots reveal deep symmetries and cyclical properties that have fascinated mathematicians for centuries. One such root, denoted (\omega), satisfies the quadratic equation (x^2 - x + 1 = 0). This article explores the mathematical behavior of (\omega), including its key properties: (\omega^2 = \omega - 1), (\omega^3 = -1), and the cyclical nature of its powers up to (\omega^6 = 1).", "---", "### Finding (\omega): Roots of the Quadratic Equation", "The equation (x^2 - x + 1 = 0) can be solved using the quadratic formula:", "[\nx = \frac{1 \pm \sqrt{(-1)^2 - 4(1)(1)}}{2(1)} = \frac{1 \pm \sqrt{-3}}{2} = \frac{1 \pm i\sqrt{3}}{2}\n]", "Thus, the two roots are complex numbers:\n[\n\omega = \frac{1 + i\sqrt{3}}{2}, \quad \omega^ = \frac{1 - i\sqrt{3}}{2}\n]\nThese are the primitive 6th roots of unity, lies on the unit circle in the complex plane, separated by 60 degrees.", "---", "### Key Property: (\omega^2 = \omega - 1)", "From the defining equation (x^2 = x - 1), substituting (x = \omega) yields:\n[\n\omega^2 = \omega - 1\n]\nThis relation allows reduction of higher powers of (\omega) into simpler linear forms.", "---", "### Computing (\omega^3): A Critical Insight", "Using (\omega^2 = \omega - 1), compute (\omega^3):\n[\n\omega^3 = \omega \cdot \omega^2 = \omega(\omega - 1) = \omega^2 - \omega\n]\nNow substitute (\omega^2 = \omega - 1):\n[\n\omega^3 = (\omega - 1) - \omega = -1\n]\nThus, (\omega^3 = -1), a pivotal identity revealing (\omega) has order 6 under multiplication, since ((\omega^3)^2 = (-1)^2 = 1).", "---", "### (\omega^6 = 1) and Cyclical Structure", "Since (\omega^3 = -1), squaring both sides gives:\n[\n\omega^6 = (-1)^2 = 1\n]\nThis confirms that (\omega) is a primitive 6th root of unity, meaning powers of (\omega) ciclo with period 6:\n[\n\omega, \omega^2, \omega^3 = -1, \omega^4 = -\omega, \omega^5 = -\omega^2, \omega^6 = 1\n]\nThis cyclical pattern reflects symmetry deep within complex algebraic structures and supports applications in number theory, polynomial factorization, and signal processing.", "---", "### Applications and Significance", "Roots like (\omega) are foundational in:\n- Cyclotomic Fields: Generalizing sums and symmetries of roots of unity.\n- Polynomial Factorization: Breaking non-trivial polynomials into irreducible components over (\mathbb{C}).\n- Signal Analysis: Modeling periodic phenomena through complex exponential forms.", "---", "### Summary", "Let (\omega) be a root of (x^2 - x + 1 = 0), satisfying (\omega^2 = \omega - 1). Then:\n[\n\omega^3 = -1, \quad \omega^6 = 1\n]\nDemonstrating periodicity and linking (\omega) to the 6th roots of unity. Understanding such roots unlocks deeper insights into symmetry, algebra, and beyond.", "---", "Keywords:* (\omega), roots of (x^2 - x + 1), (\omega^2 = \omega - 1), (\omega^3 = -1), (\omega^6 = 1), complex roots, cyclotomic roots, algebraic structures."]