View source for Root of unity

{{Short description|Number with an integer power equal to 1}}
{{Use American English|date=January 2019}}
{{Use dmy dates|date=January 2020}}
{{more citations needed|date=April 2012}}
[[File:One5Root.svg|thumb|right|The 5th roots of unity (blue points) in the [[complex plane]]]]

In [[mathematics]], a '''root of unity''' is any [[complex number]] that yields 1 when [[exponentiation|raised]] to some positive [[integer]] power {{mvar|n}}. Roots of unity are used in many branches of mathematics, and are especially important in [[number theory]], the theory of [[group character]]s, and the [[discrete Fourier transform]]. It is occasionally called a '''de Moivre number''' after French mathematician [[Abraham de Moivre]].

Roots of unity can be defined in any [[field (mathematics)|field]]. If the [[characteristic of a field|characteristic]] of the field is zero, the roots are complex numbers that are also [[algebraic integer]]s. For fields with a positive characteristic, the roots belong to a [[finite field]], and, [[converse (logic)|conversely]], every nonzero element of a finite field is a root of unity. Any [[algebraically closed field]] contains exactly {{mvar|n}} {{mvar|n}}th roots of unity, except when {{mvar|n}} is a multiple of the (positive) characteristic of the field.

==General definition==
[[File:visualisation_complex_number_roots.svg|thumb|Geometric representation of the 2nd to 6th root of a general complex number in polar form. For the ''n''th root of unity, set {{mvar|r}}&nbsp;=&nbsp;1 and {{mvar|&phi;}}&nbsp;=&nbsp;0. The principal root is in black.]]

An ''{{mvar|n}}th root of unity'', where {{mvar|n}} is a positive integer, is a number {{mvar|z}} satisfying the [[equation]]<ref>{{Cite book|author=Hadlock, Charles R.|author-link=Charles Robert Hadlock|title=Field Theory and Its Classical Problems, Volume 14|publisher=Cambridge University Press|year=2000|isbn=978-0-88385-032-9|pages=84–86|url=https://books.google.com/books?id=5s1p0CyafnEC&pg=PA84}}</ref><ref>{{cite book|last = Lang|first = Serge|chapter=Roots of unity|title=Algebra|publisher=Springer|year=2002|isbn=978-0-387-95385-4|pages=276–277|chapter-url=https://books.google.com/books?id=Fge-BwqhqIYC&pg=PA276}}</ref>
<math display="block">z^n = 1. </math>
Unless otherwise specified, the roots of unity may be taken to be [[complex number]]s (including the number 1, and the number −1 if {{mvar|n}} is [[parity (mathematics)|even]], which are complex with a zero [[complex number|imaginary part]]), and in this case, the {{mvar|n}}th roots of unity are<ref name="meserve">{{cite book
|last = Meserve |first = Bruce E.
|title = Fundamental Concepts of Algebra
|page = 52
|publisher = Dover Publications
|year = 1982}}</ref>
<math display="block">\exp\left(\frac{2k\pi i}{n}\right)=\cos\frac{2k\pi}{n}+i\sin\frac{2k\pi}{n},\qquad k=0,1,\dots, n-1.</math>

However, the defining equation of roots of unity is meaningful over any [[field (mathematics)|field]] (and even over any [[ring (mathematics)|ring]])  {{math|''F''}}, and this allows considering roots of unity in {{math|''F''}}. Whichever is the field {{math|''F''}}, the roots of unity in {{math|''F''}} are either complex numbers, if the [[characteristic (algebra)|characteristic]] of {{math|''F''}} 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|Root of unity modulo ''n'']] and [[Finite field]] for further details.

An {{mvar|n}}th root of unity is said to be '''{{visible anchor|primitive}}''' if it is not an {{mvar|m}}th root of unity for some smaller {{mvar|m}}, that is if<ref name="moskowitz">{{cite book
|last = Moskowitz |first= Martin A.
|year = 2003
|title = Adventure in Mathematics
|publisher = World Scientific
|url = https://books.google.com/books?id=YT2_Kqsnn9wC&pg=PA36
|page = 36|isbn= 9789812794949
}}</ref><ref name="lidl">{{cite book
|last1 = Lidl |first1 = Rudolf
|last2 = Pilz |first2 = Günter |author-link2 = Günter Pilz
|year = 1984
|title = Applied Abstract Algebra
|series = Undergraduate Texts in Mathematics
|url = https://books.google.com/books?id=irXSBwAAQBAJ&pg=PA149
|page = 149
|publisher = Springer
|doi = 10.1007/978-1-4615-6465-2
|isbn = 978-0-387-96166-8
}}</ref>

:<math>z^n=1\quad \text{and} \quad z^m \ne 1 \text{ for } m = 1, 2, 3, \ldots, n-1. </math>
If ''n'' is a [[prime number]], then all {{math|''n''}}th roots of unity, except 1, are primitive.<ref name="morandi">{{cite book
|last = Morandi | first = Patrick
|title = Field and Galois theory
| series = Graduate Texts in Mathematics
|year = 1996
| volume = 167
|url = https://books.google.com/books?id=jQ7c8Xqpqk0C&pg=PA74
|page = 74
|publisher = Springer
|isbn = 978-0-387-94753-2
|doi = 10.1007/978-1-4612-4040-2}}</ref>

In the above formula in terms of exponential and trigonometric functions, the primitive {{mvar|n}}th roots of unity are those for which {{mvar|k}} and {{mvar|n}} 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 {{slink|Finite field|Roots of unity}}. For the case of roots of unity in rings of [[modular arithmetic|modular integers]], see [[Root of unity modulo n|Root of unity modulo ''n'']].

==Elementary properties==
Every {{math|''n''}}th root of unity {{math|''z''}} is a primitive {{math|''a''}}th root of unity for some {{math|''a'' ≤ ''n''}}, which is the smallest positive integer such that {{math|1=''z''<sup>''a''</sup> = 1}}.

Any integer power of an {{math|''n''}}th root of unity is also an {{math|''n''}}th root of unity,<ref name="integer-power">{{cite book
|last = Reilly |first= Norman R.
|year = 2009
|title = Introduction to Applied Algebraic Systems
|url = https://books.google.com/books?id=q33he4hOlKcC&pg=PA137
|page = 137
|publisher= Oxford University Press
|isbn = 978-0-19-536787-4
}}</ref> as
:<math>(z^k)^n = z^{kn} = (z^n)^k = 1^k = 1.</math>
This is also true for negative exponents. In particular, the [[multiplicative inverse|reciprocal]] of an {{math|''n''}}th root of unity is its [[complex conjugate]], and is also an {{math|''n''}}th root of unity:<ref name="conjugate">{{cite book
|last = Rotman|first = Joseph J.|author-link = Joseph J. Rotman
|title = Advanced Modern Algebra
|year = 2015
|edition = 3rd
|volume = 1
|url = https://books.google.com/books?id=SugUCwAAQBAJ&pg=PA129
|page = 129
|publisher = American Mathematical Society
| isbn=9781470415549 }}</ref>
:<math>\frac{1}{z} = z^{-1} = z^{n-1} = \bar z.</math>

If {{math|''z''}} is an {{math|''n''}}th root of unity and {{math|''a'' &equiv; ''b'' (mod ''n'')}} then {{math|1=''z''<sup>''a''</sup> = ''z''<sup>''b''</sup>}}. Indeed, by the definition of [[modular arithmetic|congruence modulo ''n'']], {{math|1=''a'' = ''b'' + ''kn''}} for some integer {{math|''k''}}, and hence
:<math> z^a = z^{b+kn} = z^b z^{kn} = z^b (z^n)^k = z^b 1^k = z^b.</math>

Therefore, given a power {{math|''z''<sup>''a''</sup>}} of {{math|''z''}}, one has {{math|1=''z''<sup>''a''</sup> = ''z''<sup>''r''</sup>}}, where {{math|0 ≤ ''r'' < ''n''}} is the remainder of the [[Euclidean division]] of {{mvar|a}} by {{mvar|n}}.

Let {{math|''z''}} be a primitive {{math|''n''}}th root of unity. Then the powers {{math|''z''}}, {{math|''z''<sup>2</sup>}}, ..., {{math|''z''<sup>''n''−1</sup>}}, {{math|1=''z''<sup>''n''</sup> = ''z''<sup>0</sup> = 1}} are {{math|''n''}}th roots of unity and are all distinct. (If {{math|1=''z''<sup>''a''</sup> = ''z''<sup>''b''</sup>}} where {{math|1 ≤ ''a'' < ''b'' ≤ ''n''}}, then {{math|1=''z''<sup>''b''−''a''</sup> = 1}}, which would imply that {{math|''z''}} would not be primitive.) This implies that {{math|''z''}}, {{math|''z''<sup>2</sup>}}, ..., {{math|''z''<sup>''n''−1</sup>}}, {{math|1=''z''<sup>''n''</sup> = ''z''<sup>0</sup> = 1}} are all of the {{math|''n''}}th roots of unity, since an {{math|''n''}}th-[[degree of a polynomial|degree]] [[polynomial equation]] over a field (in this case the field of complex numbers) has at most {{math|''n''}} solutions.

From the preceding, it follows that, if {{math|''z''}} is a primitive {{math|''n''}}th root of unity, then <math>z^a = z^b</math> [[if and only if]] <math>a\equiv b \pmod{ n}.</math>
If {{math|''z''}} is not primitive then  <math>a\equiv b \pmod{ n}</math> implies <math>z^a = z^b,</math> but the converse may be false, as shown by the following example. If {{math|1=''n'' = 4}}, a non-primitive {{math|''n''}}th root of unity is {{math|1= ''z'' = −1}}, and one has <math>z^2 = z^4 = 1</math>, although <math> 2 \not\equiv 4 \pmod{4}.</math>

Let {{math|''z''}} be a primitive {{math|''n''}}th root of unity. A power {{math|1=''w'' = ''z''<sup>''k''</sup>}} of {{mvar|z}} is a primitive {{math|''a''}}th root of unity for 
:<math> a = \frac{n}{\gcd(k,n)},</math>
where <math>\gcd(k,n)</math> is the [[greatest common divisor]] of {{mvar|n}} and {{mvar|k}}. This results from the fact that {{math|''ka''}} is the smallest multiple of {{mvar|k}} that is also a multiple of {{mvar|n}}. In other words, {{math|''ka''}} is the [[least common multiple]] of {{mvar|k}} and {{mvar|n}}. Thus 
:<math>a =\frac{\operatorname{lcm}(k,n)}{k}=\frac{kn}{k\gcd(k,n)}=\frac{n}{\gcd(k,n)}.</math>

Thus, if {{math|''k''}} and {{math|''n''}} are [[coprime]], {{math|''z<sup>k</sup>''}} is also a primitive {{math|''n''}}th root of unity, and therefore there are {{math|''φ''(''n'')}} distinct primitive {{math|''n''}}th roots of unity (where {{math|''φ''}} is [[Euler's totient function]]). This implies that if {{math|''n''}} is a prime number, all the roots except {{math|+1}} are primitive.

In other words, if {{math|R(''n'')}} is the set of all {{math|''n''}}th roots of unity and {{math|P(''n'')}} is the set of primitive ones, {{math|R(''n'')}} is a [[disjoint union]] of the {{math|P(''n'')}}:

:<math>\operatorname{R}(n) = \bigcup_{d \,|\, n}\operatorname{P}(d),</math>

where the notation means that {{math|''d''}} goes through all the positive [[divisor]]s of {{math|''n''}}, including {{math|1}} and {{math|''n''}}.

Since the [[cardinality]] of {{math|R(''n'')}} is {{math|''n''}}, and that of {{math|P(''n'')}} is {{math|''φ''(''n'')}}, this demonstrates the classical formula

:<math>\sum_{d \,|\, n}\varphi(d) = n.</math>

==Group properties==
===Group of all roots of unity===
The product and the [[multiplicative inverse]] of two roots of unity are also roots of unity. In fact, if {{math|''x<sup>m</sup>'' {{=}} 1}} and {{math|''y<sup>n</sup>'' {{=}} 1}}, then {{math|(''x''<sup>−1</sup>){{sup|''m''}} {{=}} 1}}, and {{math|(''xy''){{sup|''k''}} {{=}} 1}}, where {{math|''k''}} is the [[least common multiple]] of {{math|''m''}} and {{math|''n''}}.

Therefore, the roots of unity form an [[abelian group]] under multiplication. This [[group (mathematics)|group]] is the [[torsion subgroup]] of the [[circle group]].

===Group of {{math|''n''}}th roots of unity===
For an integer ''n'', the product and the multiplicative inverse of two {{math|''n''}}th roots of unity are also {{math|''n''}}th roots of unity. Therefore, the {{math|''n''}}th roots of unity form an abelian group under multiplication.

Given a primitive {{math|''n''}}th root of unity {{math|''ω''}}, the other {{math|''n''}}th roots are powers of {{math|''ω''}}. This means that the group of the {{math|''n''}}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 {{math|''n''}}th roots of unity===
Let <math>\Q(\omega)</math> be the [[field extension]] of the [[rational number]]s generated over <math>\Q</math> by a primitive {{math|''n''}}th root of unity {{math|''ω''}}. As every {{math|''n''}}th root of unity is a power of {{math|''ω''}}, the [[field (mathematics)|field]] <math>\Q(\omega)</math> contains all {{math|''n''}}th roots of unity, and <math>\Q(\omega)</math> is a [[Galois extension]] of <math>\Q.</math>

If {{math|''k''}} is an integer, {{math|''ω<sup>k</sup>''}} is a primitive {{math|''n''}}th root of unity if and only if {{math|''k''}} and {{math|''n''}} are [[coprime]]. In this case, the map

:<math>\omega \mapsto \omega^k</math>

induces an [[field automorphism|automorphism]] of <math>\Q(\omega)</math>, which maps every {{math|''n''}}th root of unity to its {{math|''k''}}th power. Every automorphism of <math>\Q(\omega)</math> is obtained in this way, and these automorphisms form the [[Galois group]] of <math>\Q(\omega)</math> over the field of the rationals.

The rules of exponentiation imply that the [[function composition|composition]] of two such automorphisms is obtained by multiplying the exponents. It follows that the map

:<math>k\mapsto \left(\omega \mapsto \omega^k\right)</math>

defines a [[group isomorphism]] between the [[unit (ring theory)|units]] of the ring of [[integers modulo n|integers modulo {{math|''n''}}]] and the Galois group of <math>\Q(\omega).</math>

This shows that this Galois group is [[abelian group|abelian]], and implies thus that the primitive roots of unity may be expressed in terms of [[radical expression|radicals]].

===Galois group of the real part of the primitive roots of unity===
{{main|Minimal polynomial of 2cos(2pi/n)}}
The real part of the primitive roots of unity are related to one another as roots of the [[minimal polynomial (field theory)|minimal polynomial]] of <math>2\cos(2\pi/n).</math> The roots of the minimal polynomial are just twice the real part; these roots form a cyclic Galois group.

==Trigonometric expression==
[[Image:3rd roots of unity correction.svg|thumb|right|The cube roots of unity]]

[[De Moivre's formula]], which is valid for all [[real number|real]] {{mvar|x}} and integers {{mvar|n}}, is

:<math>\left(\cos x + i \sin x\right)^n = \cos nx + i \sin nx.</math>

Setting {{math|1=''x'' = {{sfrac|2π|''n''}}}} gives a primitive {{mvar|n}}th root of unity – one gets

:<math>\left(\cos\frac{2\pi}{n} + i \sin\frac{2\pi}{n}\right)^{\!n} = \cos 2\pi + i \sin 2\pi = 1,</math>
but
:<math>\left(\cos\frac{2\pi}{n} + i \sin\frac{2\pi}{n}\right)^{\!k} = \cos\frac{2k\pi}{n} + i \sin\frac{2k\pi}{n} \neq 1</math>
for {{math|1=''k'' = 1, 2, …, ''n'' − 1}}. In other words, 
:<math>\cos\frac{2\pi}{n} + i \sin\frac{2\pi}{n}</math>
is a primitive {{mvar|n}}th root of unity.

This formula shows that in the [[complex plane]] the {{mvar|n}}th roots of unity are at the vertices of a [[regular polygon|regular {{mvar|n}}-sided polygon]] inscribed in the [[unit circle]], with one vertex at 1 (see the plot for {{math|1=''n'' = 3}} 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 "[[wikt:κύκλος|cyclo]]" (circle) plus "[[wikt:τόμος|tomos]]" (cut, divide).

[[Euler's formula]]

:<math>e^{i x} = \cos x + i \sin x,</math>

which is valid for all real {{mvar|x}}, can be used to put the formula for the {{mvar|n}}th roots of unity into the form

:<math>e^{2\pi i \frac{k}{n}}, \quad 0 \le k < n.</math>

It follows from the discussion in the previous section that this is a primitive {{mvar|n}}th-root if and only if the fraction {{math|{{sfrac|''k''|''n''}}}} is in lowest terms; that is, that {{mvar|k}} and {{mvar|n}} are coprime.  An [[irrational number]] that can be expressed as the [[complex number|real part]] of the root of unity; that is, as <math>\cos(2\pi k/n)</math>, is called a [[trigonometric number]].

==Algebraic expression==

The {{math|''n''}}th roots of unity are, by definition, the [[root of a polynomial|roots]] of the [[polynomial]] {{math|''x<sup>n</sup>'' − 1}}, and are thus [[algebraic number]]s. As this polynomial is not [[irreducible polynomial|irreducible]] (except for {{math|1=''n'' = 1}}), the primitive {{math|''n''}}th roots of unity are roots of an irreducible polynomial (over the integers) of lower degree, called the {{math|''n''}}th [[cyclotomic polynomial]], and often denoted {{math|Φ<sub>''n''</sub>}}. The degree of {{math|Φ<sub>''n''</sub>}} is given by [[Euler's totient function]], which counts (among other things) the number of primitive {{math|''n''}}th roots of unity.<ref name="riesel">{{cite book
|last= Riesel |first= Hans |author-link= Hans Riesel
|year= 1994
|title= Prime Factorization and Computer Methods for Factorization
|url= https://books.google.com/books?id=ITvaBwAAQBAJ&pg=PA306
|page= 306
|publisher= Springer
|isbn= 0-8176-3743-5}}</ref> The roots of {{math|Φ<sub>''n''</sub>}} are exactly the primitive {{math|''n''}}th roots of unity.

[[Galois theory]] can be used to show that the cyclotomic polynomials may be conveniently solved in terms of radicals. (The trivial form <math>\sqrt[n]{1}</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 {{mvar|n}}, there exists an expression built from integers by root extractions, additions, subtractions, multiplications, and divisions (and nothing else), such that the primitive {{mvar|n}}th roots of unity are exactly the set of values that can be obtained by choosing values for the root extractions ({{mvar|k}} possible values for a {{mvar|k}}th root). (For more details see {{slink||Cyclotomic fields}}, below.)

Gauss [[mathematical proof|proved]] that a primitive {{mvar|n}}th root of unity can be expressed using only [[square root]]s, addition, subtraction, multiplication and division if and only if it is possible to [[Compass-and-straightedge construction|construct with compass and straightedge]] the [[regular polygon|regular {{mvar|n}}-gon]]. This is the case [[if and only if]] {{math|''n''}} is either a [[power of two]] or the product of a power of two and [[Fermat prime]]s that are all different.

If {{mvar|z}} is a primitive {{mvar|n}}th root of unity, the same is true for {{math|1/''z''}}, and <math>r=z+\frac 1z</math> is twice the real part of {{mvar|z}}. In other words, {{math|Φ<sub>''n''</sub>}} is a [[reciprocal polynomial]], the polynomial <math>R_n</math> that has {{mvar|r}} as a root may be deduced from {{math|Φ<sub>''n''</sub>}} by the standard manipulation on reciprocal polynomials, and the primitive {{mvar|n}}th roots of unity may be deduced from the roots of <math>R_n</math> by solving the [[quadratic equation]] <math>z^2-rz+1=0.</math> That is, the real part of the primitive root is <math>\frac r2,</math> and its imaginary part is <math>\pm i\sqrt{1-\left(\frac r2\right)^2}.</math>

The polynomial <math>R_n</math> is an irreducible polynomial whose roots are all real. Its degree is a power of two, if and only if {{mvar|n}} is a product of a power of two by a product (possibly [[empty product|empty]]) of distinct Fermat primes, and the regular {{mvar|n}}-gon is constructible with compass and straightedge. Otherwise, it is solvable in radicals, but one are in the [[casus irreducibilis]], that is, every expression of the roots in terms of radicals involves ''nonreal radicals''.

===Explicit expressions in low degrees===
* For {{math|1=''n'' = 1}}, the cyclotomic polynomial is {{math|Φ<sub>1</sub>(''x'') {{=}} ''x'' − 1}} Therefore, the only primitive first root of unity is 1, which is a non-primitive {{math|''n''}}th root of unity for every ''n'' > 1.
* As {{math|Φ<sub>2</sub>(''x'') {{=}} ''x'' + 1}}, the only primitive second (square) root of unity is −1, which is also a non-primitive {{math|''n''}}th root of unity for every even {{math|''n'' > 2}}. With the preceding case, this completes the list of [[real number|real]] roots of unity.
* As {{math|Φ<sub>3</sub>(''x'') {{=}} ''x''<sup>2</sup> + ''x'' + 1}}, the primitive third ([[cube root|cube]]) roots of unity, which are the roots of this [[quadratic polynomial]], are <math display="block">\frac{-1 + i \sqrt{3}}{2},\ \frac{-1 - i \sqrt{3}}{2} .</math>
* As {{math|Φ<sub>4</sub>(''x'') {{=}} ''x''<sup>2</sup> + 1}}, the two primitive fourth roots of unity are {{math|''i''}} and {{math|−''i''}}.
* As {{math|Φ<sub>5</sub>(''x'') {{=}} ''x''<sup>4</sup> + ''x''<sup>3</sup> + ''x''<sup>2</sup> + ''x'' + 1}}, the four primitive fifth roots of unity are the roots of this [[quartic polynomial]], which may be explicitly solved in terms of radicals, giving the roots <math display="block">\frac{\varepsilon\sqrt 5 - 1}4 \pm i \frac{\sqrt{10 + 2\varepsilon\sqrt 5}}{4},</math> where <math>\varepsilon</math> may take the two values 1 and −1 (the same value in the two occurrences).
* As {{math|Φ<sub>6</sub>(''x'') {{=}} ''x''<sup>2</sup> − ''x'' + 1}}, there are two primitive sixth roots of unity, which are the negatives (and also the square roots) of the two primitive cube roots: <math display="block">\frac{1 + i \sqrt{3}}{2},\ \frac{1 - i \sqrt{3}}{2}.</math>
* As 7 is not a Fermat prime, the seventh roots of unity are the first that require [[cube root]]s. There are 6 primitive seventh roots of unity, which are pairwise [[complex conjugate]]. 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 <math>r^3+r^2-2r-1,</math> and the primitive seventh roots of unity are <math display="block">\frac{r}{2}\pm i\sqrt{1-\frac{r^2}{4}},</math> where {{mvar|r}} 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]], and any such expression involves non-real cube roots.
* As {{math|Φ<sub>8</sub>(''x'') {{=}} ''x''<sup>4</sup> + 1}}, the four primitive eighth roots of unity are the square roots of the primitive fourth roots, {{math|±&hairsp;''i''}}. They are thus <math display="block"> \pm\frac{\sqrt{2}}{2} \pm i\frac{\sqrt{2}}{2}.</math>
* See [[Heptadecagon]] for the real part of a 17th root of unity.

==Periodicity==
If {{mvar|z}} is a primitive {{mvar|n}}th root of unity, then the sequence of powers
: {{math|… , ''z''<sup>−1</sup>, ''z''<sup>0</sup>, ''z''<sup>1</sup>, …}}
is {{mvar|n}}-periodic (because {{math|1=''z''<sup>'' j'' + ''n''</sup> = ''z''<sup>'' j''</sup>''z''<sup>'' n''</sup> = ''z''<sup>'' j''</sup>}} for all values of {{mvar|j}}), and the {{mvar|n}} sequences of powers
:{{math|''s<sub>k</sub>'': … , ''z''<sup>'' k''⋅(−1)</sup>, ''z''<sup>'' k''⋅0</sup>, ''z''<sup>'' k''⋅1</sup>, …}}
for {{math|1=''k'' = 1, … , ''n''}} are all {{mvar|n}}-periodic (because {{math|1=''z''<sup>'' k''⋅(''j'' + ''n'')</sup> = ''z''<sup>'' k''⋅''j''</sup>}}). Furthermore, the set {{math|{''s''<sub>1</sub>, … , ''s<sub>n</sub>''}}} of these sequences is a [[basis (linear algebra)|basis]] of the [[linear space]] of all {{mvar|n}}-periodic sequences. This means that ''any'' {{mvar|n}}-periodic sequence of complex numbers
: {{math|… , ''x''<sub>−1</sub> , ''x''<sub>0</sub> , ''x''<sub>1</sub>, …}}
can be expressed as a [[linear combination]] of powers of a primitive {{mvar|n}}th root of unity:
:<math> x_j = \sum_k X_k \cdot z^{k \cdot j} = X_1 z^{1\cdot j} + \cdots + X_n \cdot z^{n \cdot j}</math>
for some complex numbers {{math|1=''X''<sub>1</sub>, … , ''X''<sub>''n''</sub>}} and every integer {{mvar|j}}.

This is a form of [[Fourier analysis]]. If {{mvar|j}} is a (discrete) time variable, then {{mvar|k}} is a [[frequency]] and {{math|''X''<sub>''k''</sub>}} is a complex [[amplitude]].

Choosing for the primitive {{mvar|n}}th root of unity
:<math>z = e^\frac{2\pi i}{n} = \cos\frac{2\pi}{n} + i \sin\frac{2\pi}{n}</math>
allows {{math|''x''<sub>''j''</sub>}} to be expressed as a linear combination of {{math|cos}} and {{math|sin}}:
:<math>x_j = \sum_k A_k \cos \frac{2\pi jk}{n} + \sum_k B_k \sin \frac{2\pi jk}{n}.</math>
This is a [[discrete Fourier transform]].

==Summation==
Let {{math|SR(''n'')}} be the sum of all the {{mvar|n}}th roots of unity, primitive or not. Then

:<math>\operatorname{SR}(n) =
\begin{cases}
1, & n=1\\
0, & n>1.
\end{cases}</math>

This is an immediate consequence of [[Vieta's formulas]]. In fact, the {{mvar|n}}th roots of unity being the roots of the polynomial {{math|''X''<sup>&thinsp;''n''</sup> − 1}}, their sum is the [[coefficient]] of degree {{math|''n'' − 1}}, which is either 1 or 0 according whether {{math|1=''n'' = 1}} or {{math|''n'' > 1}}.

Alternatively, for {{math|1=''n'' = 1}} there is nothing to prove, and for {{math|1=''n'' > 1}} there exists a root {{math|''z'' ≠ 1}} – since the set {{math|''S''}} of all the {{mvar|n}}th roots of unity is a [[group (mathematics)|group]], {{math|1=''z{{space|hair}}S'' =  ''S''}}, so the sum satisfies {{math|1= ''z'' SR(''n'') =  SR(''n'')}}, whence {{math|1=SR(''n'') = 0}}.

Let {{math|SP(''n'')}} be the sum of all the primitive {{mvar|n}}th roots of unity. Then

:<math>\operatorname{SP}(n) = \mu(n),</math>

where {{math|''μ''(''n'')}} is the [[Möbius function]].

In the section [[Root of unity#Elementary properties|Elementary properties]], it was shown that if {{math|R(''n'')}} is the set of all {{mvar|n}}th roots of unity and {{math|P(''n'')}} is the set of primitive ones, {{math|R(''n'')}} is a disjoint union of the {{math|P(''n'')}}:

:<math>\operatorname{R}(n) = \bigcup_{d \,|\, n}\operatorname{P}(d),</math>

This implies

:<math>\operatorname{SR}(n) = \sum_{d \,|\, n}\operatorname{SP}(d).</math>

Applying the [[Möbius inversion formula]] gives

:<math>\operatorname{SP}(n) = \sum_{d \,|\, n}\mu(d)\operatorname{SR}\left(\frac{n}{d}\right).</math>

In this formula, if {{math|''d'' < ''n''}}, then {{math|1=SR({{sfrac|''n''|''d''}}) = 0}}, and for {{math|1=''d'' = ''n''}}: {{math|1=SR({{sfrac|''n''|''d''}}) = 1}}.  Therefore, {{math|1=SP(''n'') = ''μ''(''n'')}}.

This is the special case {{math|''c''<sub>''n''</sub>(1)}} of [[Ramanujan's sum]] {{math|''c''<sub>''n''</sub>(''s'')}},<ref name="apostol">{{cite book
|last = Apostol |first = Tom M. |author-link = Tom M. Apostol
|year = 1976
|title = Introduction to Analytic Number Theory
|series = Undergraduate Texts in Mathematics |url = https://books.google.com/books?id=3yoBCAAAQBAJ&pg=PA160
|page = 160
|publisher = Springer
|doi = 10.1007/978-1-4757-5579-4|isbn = 978-1-4419-2805-4 }}</ref> defined as the sum of the {{mvar|s}}th powers of the primitive {{mvar|n}}th roots of unity:

:<math>c_n(s) = \sum_{a = 1 \atop \gcd(a, n) = 1}^n e^{2 \pi i \frac{a}{n} s}.</math>

==Orthogonality==
From the summation formula follows an [[orthogonality]] relationship: for {{math|1=''j''&nbsp;=&thinsp;1, … , ''n''}} and {{math|1=''j′''&nbsp;=&thinsp;1, … , ''n''}}

:<math>\sum_{k=1}^{n} \overline{z^{j\cdot k}} \cdot z^{j'\cdot k} = n \cdot\delta_{j,j'}</math>

where {{mvar|&delta;}} is the [[Kronecker delta]] and {{mvar|z}} is any primitive {{mvar|n}}th root of unity.

The {{math|''n''&thinsp;×&thinsp;''n''}} [[matrix (mathematics)|matrix]] {{mvar|U}} whose {{math|(''j'', ''k'')}}th entry is

:<math>U_{j,k} = n^{-\frac{1}{2}}\cdot z^{j\cdot k}</math>

defines a [[discrete Fourier transform]].  Computing the inverse transformation using [[Gaussian elimination]] requires {{math|''[[big-O notation|O]]''(''n''<sup>3</sup>)}} operations.  However, it follows from the orthogonality that {{mvar|U}} is [[unitary matrix|unitary]].  That is,

:<math>\sum_{k=1}^{n} \overline{U_{j,k}} \cdot U_{k,j'} = \delta_{j,j'},</math>

and thus the inverse of {{mvar|U}} is simply the complex conjugate. (This fact was first noted by [[Carl Friedrich Gauss|Gauss]] when solving the problem of [[trigonometric interpolation]].)  The straightforward application of {{mvar|U}} or its inverse to a given vector requires {{math|''O''(''n''<sup>2</sup>)}} operations. The [[fast Fourier transform]] algorithms reduces the number of operations further to {{math|''O''(''n'' log ''n'')}}.

==Cyclotomic polynomials==<!-- This section is linked from [[Cyclotomic polynomial]] -->
{{Main|Cyclotomic polynomial}}

The [[zero of a function|zeros]] of the polynomial
:<math>p(z) = z^n - 1</math>
are precisely the {{mvar|n}}th roots of unity, each with [[multiplicity of a root|multiplicity]] 1. The {{mvar|n}}th ''[[cyclotomic polynomial]]'' is defined by the fact that its zeros are precisely the ''primitive'' {{mvar|n}}th roots of unity, each with multiplicity 1.
: <math>\Phi_n(z) = \prod_{k=1}^{\varphi(n)}(z-z_k)</math>
where {{math|''z''<sub>1</sub>, ''z''<sub>2</sub>, ''z''<sub>3</sub>, …, ''z''<sub>φ(''n'')</sub>}} are the primitive {{mvar|n}}th roots of unity, and {{math|φ(''n'')}} is [[Euler's totient function]]. The polynomial {{math|Φ<sub>''n''</sub>(''z'')}} 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).<ref name="riesel" /> The case of prime {{mvar|n}}, which is easier than the general assertion, follows by applying [[Eisenstein's criterion]] to the polynomial

:<math>\frac{(z+1)^n - 1}{(z+1) - 1},</math>

and expanding via the [[binomial theorem]].

Every {{mvar|n}}th root of unity is a primitive {{mvar|d}}th root of unity for exactly one positive [[divisor]] {{mvar|d}} of {{mvar|n}}. This implies that<ref name="riesel" />

:<math>z^n - 1 = \prod_{d \,|\, n} \Phi_d(z).</math>

This formula represents the [[factorization of polynomials|factorization]] of the polynomial {{math|''z''<sup>''n''</sup> − 1}} into irreducible factors:

:<math>\begin{align}
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{align}</math>

Applying [[Möbius inversion]] to the formula gives

:<math>\Phi_n(z) = \prod_{d \,|\, n}\left(z^\frac{n}{d} - 1\right)^{\mu(d)} = \prod_{d \,|\, n}\left(z^d - 1\right)^{\mu\left(\frac{n}{d}\right)},</math>

where {{math|''μ''}} is the [[Möbius function]]. So the first few cyclotomic polynomials are

:{{math|1=Φ<sub>1</sub>(''z'') = ''z'' − 1}}
:{{math|1=Φ<sub>2</sub>(''z'') = (''z''<sup>2</sup> − 1)⋅(''z'' − 1)<sup>−1</sup> = ''z'' + 1}}
:{{math|1=Φ<sub>3</sub>(''z'') = (''z''<sup>3</sup> − 1)⋅(''z'' − 1)<sup>−1</sup> = ''z''<sup>2</sup> + ''z'' + 1}}
:{{math|1=Φ<sub>4</sub>(''z'') = (''z''<sup>4</sup> − 1)⋅(''z''<sup>2</sup> − 1)<sup>−1</sup> = ''z''<sup>2</sup> + 1}}
:{{math|1=Φ<sub>5</sub>(''z'') = (''z''<sup>5</sup> − 1)⋅(''z'' − 1)<sup>−1</sup> = ''z''<sup>4</sup> + ''z''<sup>3</sup> + ''z''<sup>2</sup> + ''z'' + 1}}
:{{math|1=Φ<sub>6</sub>(''z'') = (''z''<sup>6</sup> − 1)⋅(''z''<sup>3</sup> − 1)<sup>−1</sup>⋅(''z''<sup>2</sup> − 1)<sup>−1</sup>⋅(''z'' − 1)  = ''z''<sup>2</sup> − ''z'' + 1}}
:{{math|1=Φ<sub>7</sub>(''z'') = (''z''<sup>7</sup> − 1)⋅(''z'' − 1)<sup>−1</sup>  = ''z''<sup>6</sup> + ''z''<sup>5</sup> + ''z''<sup>4</sup> + ''z''<sup>3</sup> + ''z''<sup>2</sup>  +''z'' + 1}}
:{{math|1=Φ<sub>8</sub>(''z'') = (''z''<sup>8</sup> − 1)⋅(''z''<sup>4</sup> − 1)<sup>−1</sup> = ''z''<sup>4</sup> + 1}}

If {{mvar|p}} is a [[prime number]], then all the {{mvar|p}}th roots of unity except 1 are primitive {{mvar|p}}th roots. Therefore,<ref name="morandi" />
<math display="block">\Phi_p(z) = \frac{z^p - 1}{z - 1} = \sum_{k = 0}^{p - 1} z^k.</math>
Substituting any positive integer ≥&thinsp;2 for {{mvar|z}}, this sum becomes a [[radix|base {{mvar|z}}]] [[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 {{math|Φ<sub>{{num|105}}</sub>}}. It is not a surprise it takes this long to get an example, because the behavior of the coefficients depends not so much on {{mvar|n}} as on how many [[parity (mathematics)|odd]] prime factors appear in {{mvar|n}}.  More precisely, it can be shown that if {{mvar|n}} has 1 or 2 odd prime factors (for example, {{math|1=''n''&nbsp;=&thinsp;150}}) then the {{mvar|n}}th cyclotomic polynomial only has coefficients 0, 1 or −1.  Thus the first conceivable {{mvar|n}} for which there could be a coefficient besides 0, 1, or −1 is a product of the three smallest odd primes, and that is {{math|1=3&thinsp;⋅&thinsp;5&thinsp;⋅&thinsp;7 = 105}}.  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 <math>n = p_1 p_2 \cdots p_t,</math> where <math>p_1 < p_2 < \cdots < p_t</math> are odd primes, <math>p_1 +p_2>p_t,</math> and ''t'' is odd, then {{math|1 − ''t''}} occurs as a coefficient in the {{mvar|n}}th cyclotomic polynomial.<ref name="lehmer">{{cite journal
|last = Lehmer|first = Emma|author-link = Emma Lehmer
|title = On the magnitude of the coefficients of the cyclotomic polynomial
|journal = Bulletin of the American Mathematical Society
|year = 1936
|volume = 42
|issue = 6
|pages = 389–392| doi=10.1090/S0002-9904-1936-06309-3 |doi-access = free
}}</ref>

Many restrictions are known about the values that cyclotomic polynomials can assume at integer values. For example, if {{mvar|p}} is prime, then {{math|''d''&thinsp;∣&thinsp;Φ<sub>''p''</sub>(''d'')}} if and only if {{math|''d'' ≡ 1 (mod ''p'')}}.

Cyclotomic polynomials are solvable in [[radical expression|radicals]], as roots of unity are themselves radicals. Moreover, there exist more informative radical expressions for {{mvar|n}}th roots of unity with the additional property<ref name="landau">{{Cite journal
|last1 = Landau |first1 = Susan|authorlink1 = Susan Landau
|last2 = Miller |first2 = Gary L.|authorlink2 = Gary Miller (computer scientist)
|title = Solvability by radicals is in polynomial time
|journal = Journal of Computer and System Sciences
|volume = 30
|issue = 2
|pages = 179–208
|year = 1985
|doi = 10.1016/0022-0000(85)90013-3
|doi-access = }}</ref> that every value of the expression obtained by choosing values of the radicals (for example, signs of square roots) is a primitive {{mvar|n}}th root of unity. This was already shown by [[Carl Friedrich Gauss|Gauss]] in 1797.<ref>{{Cite book|last=Gauss|first=Carl F.|author-link=Carl Friedrich Gauss|title=Disquisitiones Arithmeticae|pages=§§359–360|publisher=Yale University Press|year=1965|isbn=0-300-09473-6}}</ref> Efficient [[algorithm]]s exist for calculating such expressions.<ref>{{cite web|last1=Weber|first1=Andreas|last2=Keckeisen|first2=Michael|title=Solving Cyclotomic Polynomials by Radical Expressions|url=http://cg.cs.uni-bonn.de/personal-pages/weber/publications/pdf/WeberA/WeberKeckeisen99a.pdf|access-date=22 June 2007}}</ref>

==Cyclic groups==
The {{mvar|n}}th roots of unity form under multiplication a [[cyclic group]] of [[order (group theory)|order]] {{mvar|n}}, 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 {{mvar|n}}th root of unity.

The {{mvar|n}}th roots of unity form an irreducible [[group representation|representation]] of any cyclic group of order {{mvar|n}}. 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 [[eigenvector]]s 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]].<ref name="yoshitaka">{{cite book
|last1 = Inui|first1 = Teturo
|last2 = Tanabe|first2 = Yukito
|last3 = Onodera|first3 = Yoshitaka
|title = Group Theory and Its Applications in Physics
|publisher = Springer
|year = 1996}}</ref>{{page needed|date=February 2023}} In particular, if a circulant [[Hermitian matrix]] is considered (for example, a discretized one-dimensional [[Laplacian]] with periodic boundaries<ref name="siam">{{cite journal
|last = Strang |first= Gilbert |author-link = Gilbert Strang
|title = The discrete cosine transform
|url = http://epubs.siam.org/sam-bin/dbq/article/33674
|journal = SIAM Review
|volume = 41
|issue = 1
|pages = 135–147
|year = 1999|doi= 10.1137/S0036144598336745 |bibcode= 1999SIAMR..41..135S |url-access = subscription
}}</ref>), the orthogonality property immediately follows from the usual orthogonality of eigenvectors of Hermitian matrices.

==Cyclotomic fields==
{{Main|Cyclotomic field}}
By [[Field_extension#adjunction|adjoining]] a primitive {{mvar|n}}th root of unity to <math>\Q,</math> one obtains the {{mvar|n}}th [[cyclotomic field]] <math>\Q(\exp(2\pi i/n)).</math>This [[field (mathematics)|field]] contains all {{mvar|n}}th roots of unity and is the [[splitting field]] of the {{mvar|n}}th cyclotomic polynomial over <math>\Q.</math> The [[field extension]] <math>\Q(\exp(2\pi i /n))/\Q</math> has degree φ(''n'') and its [[Galois group]] is [[Natural transformation|naturally]] [[group isomorphism|isomorphic]] to the multiplicative [[group of units]] of the ring <math>\Z/n\Z.</math>

As the Galois group of <math>\Q(\exp(2\pi i /n))/\Q</math> 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]] can be written out explicitly in terms of [[Gaussian period]]s: this theory from the ''[[Disquisitiones Arithmeticae]]'' of [[Carl Friedrich Gauss|Gauss]] was published many years before Galois.<ref>The ''Disquisitiones'' was published in 1801, [[Évariste Galois|Galois]] was born in 1811, died in 1832, but wasn't published until 1846.</ref>

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==
[[Image:Roots of unity, golden ratio.svg|thumb|right|In the [[complex plane]], the red points are the fifth roots of unity, and the black points are the sums of a fifth root of unity and its complex conjugate.]]
[[Image:Star polygon 8-2.svg|thumb|right|160px|In the complex plane, the corners of the two squares are the eighth roots of unity.]]<!-- a better image needed -->
For {{math|1=''n'' = 1, 2}}, both roots of unity {{num|1}} and {{num|−1}} are [[integer]]s.

For three values of {{mvar|n}}, the roots of unity are [[quadratic integer]]s:
* For {{math|1=''n'' = 3, 6}} they are [[Eisenstein integer]]s ({{math|1=[[discriminant|''D'']] = −3}}).
* For {{math|1=''n'' = 4}} they are [[Gaussian integer]]s ({{math|1=''D'' = −1}}): see [[Imaginary unit]].

For four other values of {{mvar|n}}, the primitive roots of unity are not quadratic integers, but the sum of any root of unity with its [[complex conjugate]] (also an {{mvar|n}}th root of unity) is a quadratic integer.

For {{math|1=''n'' = 5, 10}}, none of the non-real roots of unity (which satisfy a [[quartic equation]]) is a quadratic integer, but the sum {{math|1=''z'' + ''{{overline|z}}'' = 2 [[real part|Re]] ''z''}} of each root with its complex conjugate (also a 5th root of unity) is an element of the [[ring (mathematics)|ring]] [[quadratic integer|{{math|'''Z'''[{{sfrac|1&thinsp;+&thinsp;{{sqrt|5}}|2}}]}}]] ({{math|1=''D'' = 5}}). 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 {{math|1=''n'' = 8}}, for any root of unity  {{math|''z'' + ''{{overline|z}}''}} equals to either 0, ±2, or ±[[square root of 2|{{sqrt|2}}]] ({{math|1=''D'' = 2}}).

For {{math|1=''n'' = 12}}, for any root of unity,  {{math|''z'' + ''{{overline|z}}''}} equals to either 0, ±1, ±2 or ±[[square root of 3|{{sqrt|3}}]] ({{math|1=''D'' = 3}}).

==See also==
* [[Argand system]]
* [[Circle group]], the unit complex numbers
* [[Cyclotomic field]]
* [[Group scheme of roots of unity]]
* [[Dirichlet character]]
* [[Ramanujan's sum]]
* [[Witt vector]]
* [[Teichmüller character]]

==Notes==
{{Reflist}}

==References==
* {{Lang Algebra|edition=3r}}
* {{cite web|first=James S.|last=Milne|author-link=James Milne (mathematician)|title=Algebraic Number Theory|work=Course Notes|url=http://www.jmilne.org/math|year=1998}}
* {{cite web|first=James S.|last=Milne|title=Class Field Theory|work=Course Notes|url=http://www.jmilne.org/math|year=1997}}
* {{Neukirch ANT}}
* {{Cite book|first=Jürgen|last=Neukirch|title=Class Field Theory|publisher=Springer-Verlag|location=Berlin|year=1986|isbn=3-540-15251-2}}
* {{Cite book|first=Lawrence C.|last=Washington|author-link=Lawrence C. Washington|title=Introduction to Cyclotomic Fields|publisher=Springer-Verlag|location=New York|year=1997|edition=2nd|isbn=0-387-94762-0}}
* {{Cite book|author=Derbyshire, John|author-link=John Derbyshire|title=Unknown Quantity|chapter=Roots of Unity|year=2006|publisher=[[Joseph Henry Press]]|location=Washington, D.C.|isbn=0-309-09657-X|url=https://archive.org/details/isbn_9780309096577|url-access=registration}}

{{Algebraic numbers}}

{{DEFAULTSORT:Root of Unity}}
[[Category:Algebraic numbers]]
[[Category:Cyclotomic fields]]
[[Category:Polynomials]]
[[Category:1 (number)]]
[[Category:Complex numbers]]