Fix K N and n N2 Show that all squareroots of 1 zk z2k

Fix K N and n N_2. Show that all squareroots of 1 + z^k + z^2k + ... + z^nk = 0 have absolute value 1.

Solution

let x = z^k

equation becomes

1 +x + ....+ x^n = 0                                                 ---- 1

(x^n+1 -1 )/(x-1) = 0         

hence the solutions of eq 1 are the solutions of x^n+1 -1 =0 except x = 1

x^{n+1} -1 = 0

so |x| = 1

now z^k = x

hence |z| = 1