In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a root of unity, occasionally called a
de Moivre
Abraham de Moivre FRS (; 26 May 166727 November 1754) was a French mathematician known for de Moivre's formula, a formula that links complex numbers and trigonometry, and for his work on the normal distribution and probability theory.
He mov ...
number, is any
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
that yields 1 when
raised to some positive
integer power . Roots of unity are used in many branches of mathematics, and are especially important in
number theory, the theory of
group characters, and the
discrete Fourier transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a comple ...
.
Roots of unity can be defined in any
field. If the
characteristic of the field is zero, the roots are complex numbers that are also
algebraic integers. For fields with a positive characteristic, the roots belong to a
finite field, and,
conversely
In logic and mathematics, the converse of a categorical or implicational statement is the result of reversing its two constituent statements. For the implication ''P'' → ''Q'', the converse is ''Q'' → ''P''. For the categorical proposit ...
, every nonzero element of a finite field is a root of unity. Any
algebraically closed field contains exactly th roots of unity, except when is a multiple of the (positive) characteristic of the field.
General definition
An ''th root of unity'', where is a positive integer, is a number satisfying the
equation
:
Unless otherwise specified, the roots of unity may be taken to be
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s (including the number 1, and the number −1 if is
even
Even may refer to:
General
* Even (given name), a Norwegian male personal name
* Even (surname)
* Even (people), an ethnic group from Siberia and Russian Far East
**Even language, a language spoken by the Evens
* Odd and Even, a solitaire game wh ...
, which are complex with a zero
imaginary part), and in this case, the th roots of unity are
:
However, the defining equation of roots of unity is meaningful over any
field (and even over any
ring) , and this allows considering roots of unity in . Whichever is the field , the roots of unity in are either complex numbers, if the
characteristic of is 0, or, otherwise, belong to a
finite field. Conversely, every nonzero element in a finite field is a root of unity in that field. See
Root of unity modulo ''n'' and
Finite field for further details.
An th root of unity is said to be if it is not an th root of unity for some smaller , that is if
:
If ''n'' is a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
, then all th roots of unity, except 1, are primitive.
In the above formula in terms of exponential and trigonometric functions, the primitive th roots of unity are those for which and are
coprime integers.
Subsequent sections of this article will comply with complex roots of unity. For the case of roots of unity in fields of nonzero characteristic, see . For the case of roots of unity in rings of
modular integers, see
Root of unity modulo ''n''.
Elementary properties
Every th root of unity is a primitive th root of unity for some , which is the smallest positive integer such that .
Any integer power of an th root of unity is also an th root of unity, as
:
This is also true for negative exponents. In particular, the
reciprocal
Reciprocal may refer to:
In mathematics
* Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal''
* Reciprocal polynomial, a polynomial obtained from another pol ...
of an th root of unity is its
complex conjugate, and is also an th root of unity:
:
If is an th root of unity and then . Indeed, by the definition of
congruence modulo ''n'', for some integer , and hence
:
Therefore, given a power of , one has , where is the remainder of the
Euclidean division of by .
Let be a primitive th root of unity. Then the powers , , ..., , are th roots of unity and are all distinct. (If where , then , which would imply that would not be primitive.) This implies that , , ..., , are all of the th roots of unity, since an th-
degree polynomial equation over a field (in this case the field of complex numbers) has at most solutions.
From the preceding, it follows that, if is a primitive th root of unity, then
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bic ...
If is not primitive then
implies
but the converse may be false, as shown by the following example. If , a non-primitive th root of unity is , and one has
, although
Let be a primitive th root of unity. A power of is a primitive th root of unity for
:
where
is the
greatest common divisor of and . This results from the fact that is the smallest multiple of that is also a multiple of . In other words, is the
least common multiple of and . Thus
:
Thus, if and are
coprime, is also a primitive th root of unity, and therefore there are distinct primitive th roots of unity (where is
Euler's totient function). This implies that if is a prime number, all the roots except are primitive.
In other words, if is the set of all th roots of unity and is the set of primitive ones, is a
disjoint union of the :
:
where the notation means that goes through all the positive
divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
s of , including and .
Since the
cardinality of is , and that of is , this demonstrates the classical formula
:
Group properties
Group of all roots of unity
The product and the
multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/' ...
of two roots of unity are also roots of unity. In fact, if and , then , and , where is the
least common multiple of and .
Therefore, the roots of unity form an
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
under multiplication. This
group is the
torsion subgroup In the theory of abelian groups, the torsion subgroup ''AT'' of an abelian group ''A'' is the subgroup of ''A'' consisting of all elements that have finite order (the torsion elements of ''A''). An abelian group ''A'' is called a torsion group (or ...
of the
circle group.
Group of th roots of unity
For an integer ''n'', the product and the multiplicative inverse of two th roots of unity are also th roots of unity. Therefore, the th roots of unity form an abelian group under multiplication.
Given a primitive th root of unity , the other th roots are powers of . This means that the group of the th roots of unity is a
cyclic group. It is worth remarking that the term of ''cyclic group'' originated from the fact that this group is a
subgroup of the
circle group.
Galois group of the primitive th roots of unity
Let
be the
field extension of the
rational numbers generated over
by a primitive th root of unity . As every th root of unity is a power of , the
field contains all th roots of unity, and
is a
Galois extension of
If is an integer, is a primitive th root of unity if and only if and are
coprime. In this case, the map
:
induces an
automorphism of
, which maps every th root of unity to its th power. Every automorphism of
is obtained in this way, and these automorphisms form the
Galois group of
over the field of the rationals.
The rules of exponentiation imply that the
composition of two such automorphisms is obtained by multiplying the exponents. It follows that the map
:
defines a
group isomorphism between the
units of the ring of
integers modulo and the Galois group of
This shows that this Galois group is
abelian
Abelian may refer to:
Mathematics Group theory
* Abelian group, a group in which the binary operation is commutative
** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms
* Metabelian group, a grou ...
, and implies thus that the primitive roots of unity may be expressed in terms of
radicals
Radical may refer to:
Politics and ideology Politics
*Radical politics, the political intent of fundamental societal change
*Radicalism (historical), the Radical Movement that began in late 18th century Britain and spread to continental Europe and ...
.
Trigonometric expression
De Moivre's formula, which is valid for all
real and integers , is
:
Setting gives a primitive th root of unity – one gets
:
but
:
for . In other words,
:
is a primitive th root of unity.
This formula shows that in the
complex plane the th roots of unity are at the vertices of a
regular -sided polygon inscribed in the
unit circle, with one vertex at 1 (see the plots for and on the right). This geometric fact accounts for the term "cyclotomic" in such phrases as
cyclotomic field and
cyclotomic polynomial; it is from the Greek roots "
cyclo" (circle) plus "
tomos" (cut, divide).
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that ...
:
which is valid for all real , can be used to put the formula for the th roots of unity into the form
:
It follows from the discussion in the previous section that this is a primitive th-root if and only if the fraction is in lowest terms; that is, that and are coprime. An
irrational number that can be expressed as the
real part of the root of unity; that is, as
, is called a
trigonometric number.
Algebraic expression
The th roots of unity are, by definition, the
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...
of the
polynomial , and are thus
algebraic numbers. As this polynomial is not
irreducible (except for ), the primitive th roots of unity are roots of an irreducible polynomial of lower degree, called the th
cyclotomic polynomial, and often denoted . The degree of is given by
Euler's totient function, which counts (among other things) the number of primitive th roots of unity. The roots of are exactly the primitive th roots of unity.
Galois theory can be used to show that cyclotomic polynomials may be conveniently solved in terms of radicals. (The trivial form