Answer :
x²+x+1 = 0
x = {-1+-√(1-4*1*1)}/2
x = ω and ω²
hence α = ω and β = ω²
α^19 = ω^19 = (ω^3)^6*ω = ω (as ω^3 = 1)
β^7 = ω^14 = (ω^3)^4*ω² = ω²
equation = x² - (sum of the root)x + product of the root
= x² - (ω+ω²)x + ω*ω²
= x² + x + 1 (1+ω+ω² = 0)
x = {-1+-√(1-4*1*1)}/2
x = ω and ω²
hence α = ω and β = ω²
α^19 = ω^19 = (ω^3)^6*ω = ω (as ω^3 = 1)
β^7 = ω^14 = (ω^3)^4*ω² = ω²
equation = x² - (sum of the root)x + product of the root
= x² - (ω+ω²)x + ω*ω²
= x² + x + 1 (1+ω+ω² = 0)