root of unity

root of unity

A

root of unity

is a number

ω

such that some power

ω

n

, where

n

is a

positive integer

, equals to

1

.

Specifically, if

K

is a field, then the

n

th roots of unity in

K

are the numbers

ω

in

K

such that

ω

n

=

1

.
Equivalently, they are all the roots of the

polynomial

X

n

-

1

. No
matter what field

K

is, the polynomial can never have more than

n

roots. Clearly

1

is an example; if

n

is even, then

-

1

will also
be an example. Beyond this, the list of possibilities depends on

K

.

•

If

K

is the set of real numbers, then

1

and

-

1

are the
only possibilities.

•

If

K

is the field of the

complex numbers

, the

fundamental
theorem of algebra

assures us that the polynomial

X

n

-

1

has exactly

n

roots (counting

multiplicities

). Comparing

X

n

-

1

with its
formal derivative (

http://planetmath.org/derivativeofpolynomial

),

n

⁢

X

n

-

1

, we see that they are

coprime

, and
therefore all the roots of

X

n

-

1

are distinct. That is, there exist

n

distinct complex numbers

ω

such that

ω

n

=

1

.

If

ζ

=

e

2

⁢

π

⁢

i

/

n

=

cos

⁡

(

2

⁢

π

/

n

)

+

i

⁢

sin

⁡

(

2

⁢

π

/

n

)

, then all the

n

th roots of unity are:

ζ

k

=

e

2

⁢

π

⁢

k

⁢

i

/

n

=

cos

⁡

(

2

⁢

π

⁢

k

/

n

)

+

i

⁢

sin

⁡

(

2

⁢

π

⁢

k

/

n

)

for

k

=

1

,

2

,

…

,

n

.

If drawn on the

complex plane

, the

n

th roots of unity are the
vertices of a

regular

n

-gon centered at the origin and with a vertex
at

1

.

•

If

K

is a

finite field

having

p

a

elements, for

p

a prime,
then

every

nonzero element is a

p

a

-

1

th root of unity (in
fact this characterizes them completely; this is the role of the
Frobenius operator). For other

n

, the answer is more complicated.
For example, if

n

is

divisible

by

p

, the formal derivative of

X

n

-

1

is

n

⁢

X

n

-

1

, which is zero since the

http://planetmath.org/node/1160

characteristic

of

K

is

p

and

n

is zero modulo

p

. So one is not guaranteed that the roots of unity will be
distinct. For example, in the field of two elements,

1

=

-

1

, so there
is only one

square root

of

1

.

If an element

ω

is an

n

th root of unity but is not an

m

th
root of unity for any

0

<

m

<

n

, then

ω

is called a

n

th root of unity

. For example,
the number

ζ

defined above is a

n

th root of unity. If

ω

∈

ℂ

is a

primitive

n

th
root of unity, then all of the primitive

n

th roots of unity have the
form

ω

m

for some

m

∈

ℤ

with

gcd

⁡

(

m

,

n

)

=

1

.

The roots of unity in any field have many special relationships to one
another, some of which are true in general and some of which depend on
the field. It is upon these relationships that the various algorithms
for computing fast Fourier transforms are based.

Finally, one could ask about

similar

situations where

K

is not a
field but some more general object. Here, things are much more
complicated. For example, in the ring of endomorphisms of a

vector
space

, the

unipotent

linear transformations

are the closest analogue
to roots of unity. They still form a group, but there may be many
more of them than

n

. In a

finite group

, every element

g

has a
power

n

such that

g

n

=

1

.

Title

root of unity

Canonical name

RootOfUnity

Date of creation

2014-11-06 15:47:15

Last modified on

2014-11-06 15:47:15

Owner

alozano (2414)

Last modified by

pahio (2872)

Numerical id

17

Author

alozano (2872)

Entry type

Definition

Classification

msc 11-00

Classification

msc 11-02

Related topic

CyclotomicPolynomial

Related topic

ExamplesOfCyclotomicPolynomials

Related topic

RamanujanSum

Related topic

Unity

Related topic

CriterionForConstructibilityOfRegularPolygon

Related topic

BinomialEquation

Defines

primitive

n

th root of unity

Defines

primitive root of unity

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