z 3 − α 3 = 0 z^3-\alpha^3=0 z 3 − α 3 = 0 z 3 − α 3 = ( z − α ) ( z 2 + α z + α 2 ) z^3-\alpha^3=(z-\alpha)(z^2+\alpha z+\alpha^2) z 3 − α 3 = ( z − α ) ( z 2 + α z + α 2 )
Then
( z − α ) ( z 2 + α z + α 2 ) = 0 (z-\alpha)(z^2+\alpha z+\alpha^2)=0 ( z − α ) ( z 2 + α z + α 2 ) = 0 z_1=\alpha\ z 2 + α z + α 2 = 0 z^2+\alpha z+\alpha^2=0 z 2 + α z + α 2 = 0
z = − α ± α 2 − 4 α 2 2 = α ( − 1 ± i 3 2 ) z={-\alpha\pm\sqrt{\alpha^2-4\alpha^2}\over2}=\alpha({-1\pm i\sqrt{3}\over2}) z = 2 − α ± α 2 − 4 α 2 = α ( 2 − 1 ± i 3 ) Cube Root of Unity Value
w 1 = 1 , r e a l w_1=1,\ real w 1 = 1 , re a l
w 2 = − 1 − i 3 2 , c o m p l e x w_2={-1- i\sqrt{3}\over2}, complex w 2 = 2 − 1 − i 3 , co m pl e x
w 3 = − 1 + i 3 2 , c o m p l e x w_3={-1+ i\sqrt{3}\over2}, \ complex w 3 = 2 − 1 + i 3 , co m pl e x
z 1 = α w 1 = α ⋅ 1 = 1 ⋅ α + i ⋅ 0 z_1=\alpha w_1=\alpha\cdot1=1\cdot\alpha+i\cdot0 z 1 = α w 1 = α ⋅ 1 = 1 ⋅ α + i ⋅ 0
z 2 = α w 2 = α ⋅ − 1 − i 3 2 = − 1 2 α − i ⋅ α 3 2 z_2=\alpha w_2=\alpha\cdot{-1- i\sqrt{3}\over2}=-{1\over 2}\alpha-i\cdot{\alpha\sqrt{3}\over 2} z 2 = α w 2 = α ⋅ 2 − 1 − i 3 = − 2 1 α − i ⋅ 2 α 3
z 3 = α w 3 = α ⋅ − 1 + i 3 2 = − 1 2 α + i ⋅ α 3 2 z_3=\alpha w_3=\alpha\cdot{-1+ i\sqrt{3}\over2}=-{1\over 2}\alpha+i\cdot{\alpha\sqrt{3}\over 2} z 3 = α w 3 = α ⋅ 2 − 1 + i 3 = − 2 1 α + i ⋅ 2 α 3
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