设w是1的n次根,w=/1,试证w满足方程1+z+z^2+…+z^n-1=0

Question

设w是1的n次根,w=/1,试证w满足方程1+z+z^2+…+z^n-1=0

Answer 1

z^n=1
z^n-1=0
(z-1)(1+z+z^2+…+z^n-1)=0
w1=1
∴其余的n-1个根w必满足(1+z+z^2+…+z^n-1)=0