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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a root of unity, occasionally called a
de Moivre Abraham de Moivre (; 26 May 166727 November 1754) was a French mathematician known for de Moivre's formula In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical ...

de Moivre
number, is any
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
that yields 1 when raised to some positive
integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...
power . Roots of unity are used in many branches of mathematics, and are especially important in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

number theory
, the theory of
group character In mathematics, more specifically in group theory, the character of a group representation is a function (mathematics), function on the group (mathematics), group that associates to each group element the trace (linear algebra), trace of the corresp ...
s, and the
discrete Fourier transform In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced Sampling (signal processing), samples of a function (mathematics), function into a same-length sequence of equally-spaced samples of the discret ...
. Roots of unity can be defined in any
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
. If the
characteristic Characteristic (from the Greek word for a property, attribute or trait Trait may refer to: * Phenotypic trait in biology, which involve genes and characteristics of organisms * Trait (computer programming), a model for structuring object-oriented ...
of the field is zero, the roots are complex numbers that are also
algebraic integer In algebraic number theory Title page of the first edition of Disquisitiones Arithmeticae, one of the founding works of modern algebraic number theory. Algebraic number theory is a branch of number theory that uses the techniques of abstrac ...
s. For fields with a positive characteristic, the roots belong to a
finite field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, and, conversely, every nonzero element of a finite field is a root of unity. Any
algebraically closed field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

equation
:z^n = 1. 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 contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
s (including the number 1, and the number –1 if is even, which are complex with a zero
imaginary part In mathematics, a complex number is a number that can be expressed in the form , where and are real numbers, and is a symbol (mathematics), symbol called the imaginary unit, and satisfying the equation . Because no "real" number satisfies this ...

imaginary part
), and in this case, the th roots of unity are :\exp\left(\frac\right)=\cos\frac+i\sin\frac,\qquad k=0,1,\dots, n-1. However, the defining equation of roots of unity is meaningful over any
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
(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 Characteristic (from the Greek word for a property, attribute or trait Trait may refer to: * Phenotypic trait in biology, which involve genes and characteristics of organisms * Trait (computer programming), a model for structuring object-oriented ...
of is 0, or, otherwise, belong to a
finite field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 :z^n=1\quad \text \quad z^m \ne 1 \text m = 1, 2, 3, \ldots, n-1. 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 In number theory, two integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...
. 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 :(z^k)^n = z^ = (z^n)^k = 1^k = 1. 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 poly ...

reciprocal
of an th root of unity is its
complex conjugate In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
, and is also an th root of unity: :\frac = z^ = z^ = \bar z. If is an th root of unity and then . Indeed, by the definition of congruence modulo ''n'', for some integer , and hence : z^a = z^ = z^b z^ = z^b (z^n)^k = z^b 1^k = z^b. Therefore, given a power of , one has , where is the remainder of the
Euclidean division In arithmetic, Euclidean division – or division with remainder – is the process of division (mathematics), dividing one integer (the dividend) by another (the divisor), in a way that produces a quotient and a remainder smaller than the divisor ...
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 Degree may refer to: As a unit of measurement * Degree symbol (°), a notation used in science, engineering, and mathematics * Degree (angle), a unit of angle measurement * Degree (temperature), any of various units of temperature measurement ...
polynomial equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
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 z^a = z^b
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
a\equiv b \pmod. If is not primitive then a\equiv b \pmod implies z^a = z^b, but the converse may be false, as shown by the following example. If , a non-primitive th root of unity is , and one has z^2 = z^4 = 1, although 2 \not\equiv 4 \pmod. Let be a primitive th root of unity. A power of is a primitive th root of unity for : a = \frac, where \gcd(k,n) is the
greatest common divisor In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...

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 In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', ...

least common multiple
of and . Thus :a =\frac=\frac=\frac. Thus, if and are
coprime In number theory, two integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...
, is also a primitive th root of unity, and therefore there are distinct primitive th roots of unity (where is
Euler's totient function In number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and numbe ...
). 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

disjoint union
of the : :\operatorname(n) = \bigcup_\operatorname(d), where the notation means that goes through all the positive
divisor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

divisor
s of , including and . Since the
cardinality In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of is , and that of is , this demonstrates the classical formula :\sum_\varphi(d) = n.


Group properties


Group of all roots of unity

The product and the
multiplicative inverse Image:Hyperbola one over x.svg, thumbnail, 300px, alt=Graph showing the diagrammatic representation of limits approaching infinity, The reciprocal function: . For every ''x'' except 0, ''y'' represents its multiplicative inverse. The graph forms a r ...

multiplicative inverse
of two roots of unity are also roots of unity. In fact, if and , then , and , where is the
least common multiple In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', ...

least common multiple
of and . Therefore, the roots of unity form an
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
under multiplication. This
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
is the
torsion subgroupIn the theory of abelian group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (math ...
of the
circle group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
.


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 In group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...

cyclic group
. It is worth remarking that the term of ''cyclic group'' originated from the fact that this group is a
subgroup In group theory, a branch of mathematics, given a group (mathematics), group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely ...
of the
circle group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
.


Galois group of the primitive th roots of unity

Let \Q(\omega) be the
field extension In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
of the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s generated over \Q by a primitive th root of unity . As every th root of unity is a power of , the
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
\Q(\omega) contains all th roots of unity, and \Q(\omega) is a
Galois extension In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
of \Q. If is an integer, is a primitive th root of unity if and only if and are
coprime In number theory, two integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...
. In this case, the map :\omega \mapsto \omega^k induces an
automorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
of \Q(\omega), which maps every th root of unity to its th power. Every automorphism of \Q(\omega) is obtained in this way, and these automorphisms form the
Galois group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
of \Q(\omega) over the field of the rationals. The rules of exponentiation imply that the
composition Composition or Compositions may refer to: Arts * Composition (dance), practice and teaching of choreography * Composition (music), an original piece of music and its creation *Composition (visual arts) The term composition means "putting togethe ...
of two such automorphisms is obtained by multiplying the exponents. It follows that the map :k\mapsto \left(\omega \mapsto \omega^k\right) defines a
group isomorphism In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two gr ...
between the
units Unit may refer to: Arts and entertainment * UNIT Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in ...
of the ring of integers modulo and the Galois group of \Q(\omega). This shows that this Galois group is abelian, and implies thus that the primitive roots of unity may be expressed in terms of radicals.


Trigonometric expression

De Moivre's formula In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, which is valid for all
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
and integers , is :\left(\cos x + i \sin x\right)^n = \cos nx + i \sin nx. Setting gives a primitive th root of unity – one gets :\left(\cos\frac + i \sin\frac\right)^ = \cos 2\pi + i \sin 2\pi = 1, but :\left(\cos\frac + i \sin\frac\right)^ = \cos\frac + i \sin\frac \neq 1 for . In other words, :\cos\frac + i \sin\frac is a primitive th root of unity. This formula shows that in the
complex plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
the th roots of unity are at the vertices of a regular -sided polygon inscribed in the
unit circle In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

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 In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to , the field of rational numbers. Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of ...
and
cyclotomic polynomial In mathematics, the ''n''th cyclotomic polynomial, for any positive integer ''n'', is the unique irreducible polynomial with integer coefficients that is a divisor of x^n-1 and is not a divisor of x^k-1 for any Its root of a function, roots are al ...
; it is from the Greek roots " cyclo" (circle) plus "
tomos Tomos ( sl, link=yes, To-Tovarna, Mo-motorjev, S-Sežana, "Motorcycle Company Sežana") was a moped The term moped ( ) originally referred to a type of small motorcycle with both a motorcycle engine and bicycle pedals, generally having a les ...
" (cut, divide).
Euler's formula Euler's formula, named after Leonhard Euler Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) incl ...

Euler's formula
:e^ = \cos x + i \sin x, which is valid for all real , can be used to put the formula for the th roots of unity into the form :e^, \quad 0 \le k < n. 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
that can be expressed as the
real part In mathematics, a complex number is a number that can be expressed in the form , where and are real numbers, and is a symbol (mathematics), symbol called the imaginary unit, and satisfying the equation . Because no "real" number satisfies this ...

real part
of the root of unity; that is, as \cos(2\pi k/n), is called a
trigonometric numberIn mathematics, a trigonometric number is an irrational number produced by taking the sine or cosine In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which ...
.


Algebraic expression

The th roots of unity are, by definition, the
roots A root In vascular plant Vascular plants (from Latin ''vasculum'': duct), also known as Tracheophyta (the tracheophytes , from Greek τραχεῖα ἀρτηρία ''trācheia artēria'' 'windpipe' + φυτά ''phutá'' 'plants'), form a lar ...
of the
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

polynomial
, and are thus
algebraic number An algebraic number is any complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, ev ...
s. 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 In mathematics, the ''n''th cyclotomic polynomial, for any positive integer ''n'', is the unique irreducible polynomial with integer coefficients that is a divisor of x^n-1 and is not a divisor of x^k-1 for any Its root of a function, roots are al ...
, and often denoted . The degree of is given by
Euler's totient function In number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and numbe ...
, 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 In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems ...
can be used to show that cyclotomic polynomials may be conveniently solved in terms of radicals. (The trivial form \sqrt /math> is not convenient, because it contains non-primitive roots, such as 1, which are not roots of the cyclotomic polynomial, and because it does not give the real and imaginary parts separately.) This means that, for each positive integer , there exists an expression built from integers by root extractions, additions, subtractions, multiplications, and divisions (and nothing else), such that the primitive th roots of unity are exactly the set of values that can be obtained by choosing values for the root extractions ( possible values for a th root). (For more details see , below.) Gauss proved that a primitive th root of unity can be expressed using only
square root In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

square root
s, addition, subtraction, multiplication and division if and only if it is possible to construct with compass and straightedge the regular -gon. This is the case
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
is either a
power of two A power of two is a number of the form where is an integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Comm ...
or the product of a power of two and
Fermat prime In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s that are all different. If is a primitive th root of unity, the same is true for , and r=z+\frac 1z is twice the real part of . In other words, is a
reciprocal polynomial In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
, the polynomial R_n that has as a root may be deduced from by the standard manipulation on reciprocal polynomials, and the primitive th roots of unity may be deduced from the roots of R_n by solving the
quadratic equation In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. ...

quadratic equation
z^2-rz+1=0. That is, the real part of the primitive root is \frac r2, and its imaginary part is \pm i\sqrt. The polynomial R_n is an irreducible polynomial whose roots are all real. Its degree is a power of two, if and only if is a product of a power of two by a product (possibly empty) of distinct Fermat primes, and the regular -gon is constructible with compass and straightedge. Otherwise, it is solvable in radicals, but one are in the
casus irreducibilis In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
, that is, every expression of the roots in terms of radicals involves ''nonreal radicals''.


Explicit expressions in low degrees

*For , the cyclotomic polynomial is Therefore, the only primitive first root of unity is 1, which is a non-primitive th root of unity for every ''n'' > 1. *As , the only primitive second (square) root of unity is –1, which is also a non-primitive th root of unity for every even . With the preceding case, this completes the list of
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
roots of unity. *As , the primitive third (
cube In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

cube
) roots of unity, which are the roots of this
quadratic polynomial In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...
, are ::\frac,\ \frac . *As , the two primitive fourth roots of unity are and . *As , the four primitive fifth roots of unity are the roots of this
quartic polynomial In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis ...
, which may be explicitly solved in terms of radicals, giving the roots ::\frac4 \pm i \frac, :where \varepsilon may take the two values 1 and –1 (the same value in the two occurrences). *As , there are two primitive sixth roots of unity, which are the negatives (and also the square roots) of the two primitive cube roots: ::\frac,\ \frac. *As 7 is not a Fermat prime, the seventh roots of unity are the first that require
cube root In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

cube root
s. There are 6 primitive seventh roots of unity, which are pairwise
complex conjugate In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
. The sum of a root and its conjugate is twice its real part. These three sums are the three real roots of the cubic polynomial r^3+r^2-2r-1, and the primitive seventh roots of unity are ::\frac\pm i\sqrt, :where runs over the roots of the above polynomial. As for every cubic polynomial, these roots may be expressed in terms of square and cube roots. However, as these three roots are all real, this is
casus irreducibilis In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
, and any such expression involves non-real cube roots. *As , the four primitive eighth roots of unity are the square roots of the primitive fourth roots, . They are thus :: \pm\frac \pm i\frac. *See
Heptadecagon In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...

Heptadecagon
for the real part of a 17th root of unity.


Periodicity

If is a primitive th root of unity, then the sequence of powers : is -periodic (because for all values of ), and the sequences of powers : for are all -periodic (because ). Furthermore, the set of these sequences is a
basis Basis may refer to: Finance and accounting *Adjusted basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items. Adjusted Basis or Adjusted Tax Basis refers to the original cost or other b ...
of the
linear space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...
of all -periodic sequences. This means that ''any'' -periodic sequence of complex numbers : can be expressed as a
linear combination In mathematics, a linear combination is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' and ''y'' would be ...
of powers of a primitive th root of unity: : x_j = \sum_k X_k \cdot z^ = X_1 z^ + \cdots + X_n \cdot z^ for some complex numbers and every integer . This is a form of
Fourier analysis In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
. If is a (discrete) time variable, then is a
frequency Frequency is the number of occurrences of a repeating event per unit of time A unit of time is any particular time Time is the indefinite continued sequence, progress of existence and event (philosophy), events that occur in an apparen ...

frequency
and is a complex
amplitude The amplitude of a period Period may refer to: Common uses * Era, a length or span of time * Full stop (or period), a punctuation mark Arts, entertainment, and media * Period (music), a concept in musical composition * Period, a descriptor for ...

amplitude
. Choosing for the primitive th root of unity :z = e^\frac = \cos\frac + i \sin\frac allows to be expressed as a linear combination of and : :x_j = \sum_k A_k \cos \frac + \sum_k B_k \sin \frac. This is a
discrete Fourier transform In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced Sampling (signal processing), samples of a function (mathematics), function into a same-length sequence of equally-spaced samples of the discret ...
.


Summation

Let be the sum of all the th roots of unity, primitive or not. Then :\operatorname(n) = \begin 1, & n=1\\ 0, & n>1. \end This is an immediate consequence of
Vieta's formulas In mathematics, Vieta's formulas are formulas that relate the coefficients of a polynomial to sums and products of its Root of a function, roots. Named after François Viète (more commonly referred to by the Latinised form of his name, "Franciscu ...
. In fact, the th roots of unity being the roots of the polynomial , their sum is the
coefficient In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of degree , which is either 1 or 0 according whether or . Alternatively, for there is nothing to prove, and for there exists a root – since the set of all the th roots of unity is a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
, , so the sum satisfies , whence . Let be the sum of all the primitive th roots of unity. Then :\operatorname(n) = \mu(n), where is the
Möbius function The Möbius function is an important multiplicative function :''Outside number theory, the term multiplicative function is usually used for completely multiplicative functions. This article discusses number theoretic multiplicative functions.'' ...
. In the section Elementary properties, it was shown that if is the set of all th roots of unity and is the set of primitive ones, is a disjoint union of the : :\operatorname(n) = \bigcup_\operatorname(d), This implies :\operatorname(n) = \sum_\operatorname(d). Applying the
Möbius inversion formula In mathematics, the classic Möbius inversion formula is a relation between pairs of arithmetic functions, each defined from the other by sums over divisors. It was introduced into number theory in 1832 by August Ferdinand Möbius. A large general ...
gives :\operatorname(n) = \sum_\mu(d)\operatorname\left(\frac\right). In this formula, if , then , and for : . Therefore, . This is the special case of
Ramanujan's sum In number theory, a branch of mathematics, Ramanujan's sum, usually denoted ''cq''(''n''), is a function of two positive integer variables ''q'' and ''n'' defined by the formula: :c_q(n)= \sum_ e^, where (''a'', ''q'') = 1 means that ''a'' only t ...
, defined as the sum of the th powers of the primitive th roots of unity: :c_n(s) = \sum_^n e^.


Orthogonality

From the summation formula follows an
orthogonality In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

orthogonality
relationship: for and :\sum_^ \overline \cdot z^ = n \cdot\delta_ where is the
Kronecker delta In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
and is any primitive th root of unity. The
matrix Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols, or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the material in between a eukaryoti ...
whose th entry is :U_ = n^\cdot z^ defines a
discrete Fourier transform In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced Sampling (signal processing), samples of a function (mathematics), function into a same-length sequence of equally-spaced samples of the discret ...
. Computing the inverse transformation using Gaussian elimination requires operations. However, it follows from the orthogonality that is
unitary Unitary may refer to: * Unitary construction, in automotive design a common term for unibody (unitary body/chassis) construction * Lethal Unitary Chemical Agents and Munitions (Unitary), as chemical weapons opposite of Binary * Unitarianism, in Chr ...
. That is, :\sum_^ \overline \cdot U_ = \delta_, and thus the inverse of is simply the complex conjugate. (This fact was first noted by
Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of m ...

Gauss
when solving the problem of trigonometric interpolation). The straightforward application of or its inverse to a given vector requires operations. The
fast Fourier transform A fast Fourier transform (FFT) is an algorithm In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always and are u ...
algorithms reduces the number of operations further to .


Cyclotomic polynomials

The zeros of the polynomial :p(z) = z^n - 1 are precisely the th roots of unity, each with multiplicity 1. The th ''
cyclotomic polynomial In mathematics, the ''n''th cyclotomic polynomial, for any positive integer ''n'', is the unique irreducible polynomial with integer coefficients that is a divisor of x^n-1 and is not a divisor of x^k-1 for any Its root of a function, roots are al ...
'' is defined by the fact that its zeros are precisely the ''primitive'' th roots of unity, each with multiplicity 1. : \Phi_n(z) = \prod_^(z-z_k) where are the primitive th roots of unity, and is
Euler's totient function In number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and numbe ...
. The polynomial has integer coefficients and is an irreducible polynomial over the rational numbers (that is, it cannot be written as the product of two positive-degree polynomials with rational coefficients). The case of prime , which is easier than the general assertion, follows by applying Eisenstein's criterion to the polynomial :\frac, and expanding via the binomial theorem. Every th root of unity is a primitive th root of unity for exactly one positive
divisor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

divisor
of . This implies that :z^n - 1 = \prod_ \Phi_d(z). This formula represents the factorization of polynomials, factorization of the polynomial into irreducible factors: :\begin z^1 -1 &= z-1 \\ z^2 -1 &= (z-1)(z+1) \\ z^3 -1 &= (z-1) (z^2 + z + 1) \\ z^4 -1 &= (z-1)(z+1) (z^2+1) \\ z^5 -1 &= (z-1) (z^4 + z^3 +z^2 + z + 1) \\ z^6 -1 &= (z-1)(z+1) (z^2 + z + 1) (z^2 - z + 1)\\ z^7 -1 &= (z-1) (z^6+ z^5 + z^4 + z^3 + z^2 + z + 1) \\ z^8 -1 &= (z-1)(z+1) (z^2+1) (z^4+1) \\ \end Applying Möbius inversion to the formula gives :\Phi_n(z) = \prod_\left(z^\frac - 1\right)^ = \prod_\left(z^d - 1\right)^, where is the
Möbius function The Möbius function is an important multiplicative function :''Outside number theory, the term multiplicative function is usually used for completely multiplicative functions. This article discusses number theoretic multiplicative functions.'' ...
. So the first few cyclotomic polynomials are : : : : : : : : If 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 the th roots of unity except 1 are primitive th roots, and we have : \Phi_p(z) = \frac = \sum_^ z^k. Substituting any positive integer ≥ 2 for , this sum becomes a radix, base repunit. Thus a necessary (but not sufficient) condition for a repunit to be prime is that its length be prime. Note that, contrary to first appearances, ''not'' all coefficients of all cyclotomic polynomials are 0, 1, or −1. The first exception is . It is not a surprise it takes this long to get an example, because the behavior of the coefficients depends not so much on as on how many parity (mathematics), odd prime factors appear in . More precisely, it can be shown that if has 1 or 2 odd prime factors (for example, ) then the th cyclotomic polynomial only has coefficients 0, 1 or −1. Thus the first conceivable for which there could be a coefficient besides 0, 1, or −1 is a product of the three smallest odd primes, and that is . This by itself doesn't prove the 105th polynomial has another coefficient, but does show it is the first one which even has a chance of working (and then a computation of the coefficients shows it does). A theorem of Schur says that there are cyclotomic polynomials with coefficients arbitrarily large in absolute value. In particular, if n = p_1 p_2 \cdots p_t, where p_1 < p_2 < \cdots < p_t are odd primes, p_1 +p_2>p_t, and ''t'' is odd, then occurs as a coefficient in the th cyclotomic polynomial. Many restrictions are known about the values that cyclotomic polynomials can assume at integer values. For example, if is prime, then if and only . Cyclotomic polynomials are solvable in radicals, as roots of unity are themselves radicals. Moreover, there exist more informative radical expressions for th roots of unity with the additional property that every value of the expression obtained by choosing values of the radicals (for example, signs of square roots) is a primitive th root of unity. This was already shown by
Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of m ...

Gauss
in 1797. Efficient algorithms exist for calculating such expressions.


Cyclic groups

The th roots of unity form under multiplication a
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cyclic group
of order (group theory), order , and in fact these groups comprise all of the finite group, finite subgroups of the multiplicative group of the complex number field. A Generating set of a group, generator for this cyclic group is a primitive th root of unity. The th roots of unity form an irreducible group representation, representation of any cyclic group of order . The orthogonality relationship also follows from group theory, group-theoretic principles as described in Character group. The roots of unity appear as entries of the eigenvectors of any circulant matrix; that is, matrices that are invariant under cyclic shifts, a fact that also follows from group representation theory as a variant of Bloch's theorem. In particular, if a circulant Hermitian matrix is considered (for example, a discretized one-dimensional Laplacian with periodic boundaries), the orthogonality property immediately follows from the usual orthogonality of eigenvectors of Hermitian matrices.


Cyclotomic fields

By adjoining a primitive th root of unity to \Q, one obtains the th
cyclotomic field In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to , the field of rational numbers. Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of ...
\Q(\exp(2\pi i/n)).This
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
contains all th roots of unity and is the splitting field of the th cyclotomic polynomial over \Q. The
field extension In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
\Q(\exp(2\pi i /n))/\Q has degree φ(''n'') and its
Galois group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
is Natural transformation, naturally group isomorphism, isomorphic to the multiplicative group of units of the ring \Z/n\Z. As the Galois group of \Q(\exp(2\pi i /n))/\Q is abelian, this is an abelian extension. Every field extension, subfield of a cyclotomic field is an abelian extension of the rationals. It follows that every ''n''th root of unity may be expressed in term of ''k''-roots, with various ''k'' not exceeding φ(''n''). In these cases
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems ...
can be written out explicitly in terms of Gaussian periods: this theory from the ''Disquisitiones Arithmeticae'' of
Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of m ...

Gauss
was published many years before Galois.The ''Disquisitiones'' was published in 1801, Évariste Galois, Galois was born in 1811, died in 1832, but wasn't published until 1846. Conversely, ''every'' abelian extension of the rationals is such a subfield of a cyclotomic field – this is the content of a theorem of Leopold Kronecker, Kronecker, usually called the ''Kronecker–Weber theorem'' on the grounds that Weber completed the proof.


Relation to quadratic integers

For , both roots of unity and are
integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...
s. For three values of , the roots of unity are quadratic integers: * For they are Eisenstein integers (). * For they are Gaussian integers (): see Imaginary unit. For four other values of , the primitive roots of unity are not quadratic integers, but the sum of any root of unity with its
complex conjugate In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
(also an th root of unity) is a quadratic integer. For , none of the non-real roots of unity (which satisfy a quartic equation) is a quadratic integer, but the sum of each root with its complex conjugate (also a 5th root of unity) is an element of the ring quadratic integer, Z[] (). For two pairs of non-real 5th roots of unity these sums are multiplicative inverse, inverse golden ratio and additive inverse, minus golden ratio. For , for any root of unity equals to either 0, ±2, or ±square root of 2, (). For , for any root of unity, equals to either 0, ±1, ±2 or ±square root of 3, ().


See also

* Argand system * Circle group, the unit complex numbers * Cyclotomic field * Group scheme of roots of unity * Dirichlet character *
Ramanujan's sum In number theory, a branch of mathematics, Ramanujan's sum, usually denoted ''cq''(''n''), is a function of two positive integer variables ''q'' and ''n'' defined by the formula: :c_q(n)= \sum_ e^, where (''a'', ''q'') = 1 means that ''a'' only t ...
* Witt vector * Teichmüller character


Notes


References

* * * * * * * {{DEFAULTSORT:Root of Unity Algebraic numbers Cyclotomic fields Polynomials 1 (number) Complex numbers