Prove that `omega^4=omega^3.omega=omega` and `omega^5=omega^3.omega^2=omega^2`

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

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

If omega is a complex cube root of unity,show 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)]=[000]

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 is a non-real cube root of unity, then (1+2omega+3omega^2)/(2+3omega+omega^2)+(2+3omega+omega^2)/(3+omega+2omega^2) is equal to

If omega is an imaginary cube root of unity then prove that If omega is the cube root of unity, Find the value of (1+5omega^2+omega^4)(1+5omega^4+omega^2)(5omega^3+omega+omega^2)

Prove that (p+q omega+r omega^(2))/(r+p omega+q omega^(2))=omega^(2)

(1-omega+omega^(2))(1+omega-omega^(2))=4