Question

if `alpha` and `beta` are imaginary cube root of unity then prove `(alpha)^4 + (beta)^4 + (alpha)^-1 . (beta)^-1 = 0`

Answer 1

`1 + alpha + beta = 0`
`1* alpha*beta = 1`
`alpha^3 = beta^3 = 1`
now, `alpha^4 - beta^4 + alpha^-1beta^-1`
`alpha - beta + 1/alpha * 1/beta`
`alpha - beta + 1`
`-beta - beta = -2beta`
now `alpha^4 + beta^4 + 1/(2 beta)`
`alpha + beta + 1=0`
hence proved