If 1,α,α2,α3,............,αn−1 are roots of unity.What is the value of If 1 × α × α2 × α3, × ............,αn−1.
(−1)n−1
If 1,α,α2,α3,............,αn−1 are the roots of unity.
Then 1 × α × α2 × α3, × ............,αn−1.
= α1+2+3+.....(n−1)
=α(n−1)(n−1+1)2
=αn(n−1)2
= (cos2πn+isin2πn)n(n−1)2
= cosπ(n−1)+isinπ(n−1)
= (cosπ+isinπ)n−1
=(−1)n−1