1Complex numbers

IA Vectors and Matrices

1.3 Roots of unity

Definition

(Roots of unity)

.

The

n

th roots of unity are the roots to the equation

z

n

= 1 for

n ∈ N

. Since this is a polynomial of order

n

, there are

n

roots of

unity. In fact, the nth roots of unity are exp

2πi

k

n

for k = 0, 1, 2, 3 ···n − 1.

Proposition. If ω = exp

2πi

n

, then 1 + ω + ω

2

+ ··· + ω

n−1

= 0

Proof. Two proofs are provided:

(i)

Consider the equation

z

n

= 1. The coefficient of

z

n−1

is the sum of

all roots. Since the coefficient of

z

n−1

is 0, then the sum of all roots

= 1 + ω + ω

2

+ ··· + ω

n−1

= 0.

(ii)

Since

ω

n

−

1 = (

ω −

1)(1 +

ω

+

···

+

ω

n−1

) and

ω 6

= 1, dividing by (

ω −

1),

we have 1 + ω + ··· + ω

n−1

= (ω

n

− 1)/(ω − 1) = 0.