Simplify `|[1,1+w,1+omega^(2)],[1+w,1+omega^(2),1],[1+omega,1,1+omega]|`

|[1,omega,omega^2] , [omega, omega^2,1] , [omega^2,1,omega]|=0

det [[1, omega, omega^(2) omega, omega^(2), 1omega^(2), 1, omega]]

det [[1, omega, omega^(2) omega, omega^(2), 1omega^(2), 1, omega]] =

|[omega+omega^(2),1,omega],[omega^(2)+1,omega^(2),1],[1+omega,omega,omega^(2)]|

if omega!=1 is a complex cube root of unity, and x+iy=|[1,i,-omega],[-I,1,omega^2],[omega,-omega^2,1]|

Let omega = - (1)/(2) + i (sqrt3)/(2) , then the value of the determinant |(1,1,1),(1,-1- omega^(2),omega^(2)),(1,omega^(2),omega^(4))| , is

If omega is a cube root of unity , then |(x+1 , omega , omega^2),(omega , x+omega^2, 1),(omega^2 , 1, x+omega)| =