Reference no: EM132777129 , Length: 219 Words

Question 1: A (real or complex) scalar u is said to be an nth root of unity if uⁿ = 1.

(a) Show that if u is an nth root of unity and u ≠ . then

1 + u + + u2 + .....+ u^n-1 = 0

[Hint: 1 - uⁿ = (1 -u)(1 + u + u² + ..... + u^n-1)] 

(b) Let ω_n = e2Πi/n. Use Euler's formula (e^jθ = cosθ + isinθ) to show that ω_n is an nth root of unity.

(c) Show that if j and k are positive integers and if u_j = ω_n^j-1 and z_k = ω_n^-(k-1), then uj, zk, and ujzk are all nth roots of unity.