Let a and b be the roots of the equation x2 + x + 1 = 0. The equation whose roots are a19, b7 is:
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x² + x + 1 = 0
multiply the equation by (1 - x) => 1 - x³ = 0
x is a cube root of 1: denoted by ω or w² : as 1 + ω + ω² = 0 or 1 + ω² + ω⁴ = 0
a = ω and b = ω²
a¹⁹ = a¹⁸⁺¹ = (a³)⁶ * a = 1⁶ * a = a = ω
b⁷ = b⁶ * b = (b³)² * b = 1² * b = b = ω²
multiply the equation by (1 - x) => 1 - x³ = 0
x is a cube root of 1: denoted by ω or w² : as 1 + ω + ω² = 0 or 1 + ω² + ω⁴ = 0
a = ω and b = ω²
a¹⁹ = a¹⁸⁺¹ = (a³)⁶ * a = 1⁶ * a = a = ω
b⁷ = b⁶ * b = (b³)² * b = 1² * b = b = ω²
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