Prove that
`(1-omega^2)+(1+omega^2)^2+(1+omega)^3 = 0`

If omega is a cube root of unity, prove that (1+omega-omega^2)^3-(1-omega+omega^2)^3=0

Prove that (1-omega-omega^(2))(1-omega+omega^(2))(1+omega-omega^(2))=8

Prove that omega(1+omega-omega^(2))=-2

If omega is cube roots of unity, prove that {[(1,omega,omega^2),(omega,omega^2,1),(omega^2,1,omega)]+[(omega,omega^2,1),(omega^2,1,omega),(omega,omega^2,1)]} [(1),(omega),(omega^2)]=[(0),(0),(0)]

If omega!=1 is a complex cube root of unity, then prove that [{:(1+2omega^(2017)+omega^(2018)," "omega^(2018),1),(1,1+2omega^(2018)+omega^(2017),omega^(2017)),(omega^(2017),omega^(2018),2+2omega^(2017)+omega^(2018)):}] is singular

If omega is the complex cube root of unity,then prove that det[[1,1,11,-1-omega^(2),omega^(2)1,omega^(2),omega^(4)]]=+-3sqrt(3)i

If 1, omega, omega^2 be three roots of 1, show that: (1-omega+omega^2)^2+(1+omega-omega^2)^2=-4

If omega be an imaginary cube root of unity, show that: 1/(1+2omega)+ 1/(2+omega) - 1/(1+omega)=0 .