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.