# Root of unity

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Number with an integer power equal to 1

The 5th roots of unity (blue points) in the [complex plane](/source/Complex_plane)

In [mathematics](/source/Mathematics), a **root of unity** is any [complex number](/source/Complex_number) that yields 1 when [raised](/source/Exponentiation) to some positive [integer](/source/Integer) power n. Roots of unity are used in many branches of mathematics, and are especially important in [number theory](/source/Number_theory), the theory of [group characters](/source/Group_character), and the [discrete Fourier transform](/source/Discrete_Fourier_transform). It is occasionally called a **de Moivre number** after French mathematician [Abraham de Moivre](/source/Abraham_de_Moivre).

Roots of unity can be defined in any [field](/source/Field_(mathematics)). If the [characteristic](/source/Characteristic_of_a_field) of the field is zero, the roots are complex numbers that are also [algebraic integers](/source/Algebraic_integer). For fields with a positive characteristic, the roots belong to a [finite field](/source/Finite_field), and, [conversely](/source/Converse_(logic)), every nonzero element of a finite field is a root of unity. Any [algebraically closed field](/source/Algebraically_closed_field) contains exactly n nth roots of unity, except when n is a multiple of the (positive) characteristic of the field.

## General definition

Geometric representation of the 2nd to 6th root of a general complex number in polar form. For the *n*th root of unity, set r = 1 and φ = 0. The principal root is in black.

An *nth root of unity*, where n is a positive integer, is a number z satisfying the [equation](/source/Equation)[1][2] z n = 1. {\displaystyle z^{n}=1.} Unless otherwise specified, the roots of unity may be taken to be [complex numbers](/source/Complex_number) (including the number 1, and the number −1 if n is [even](/source/Parity_(mathematics)), which are complex with a zero [imaginary part](/source/Complex_number)), and in this case, the nth roots of unity are[3] exp ⁡ ( 2 k π i n ) = cos ⁡ 2 k π n + i sin ⁡ 2 k π n , k = 0 , 1 , … , n − 1. {\displaystyle \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.}

However, the defining equation of roots of unity is meaningful over any [field](/source/Field_(mathematics)) (and even over any [ring](/source/Ring_(mathematics))) *F*, and this allows considering roots of unity in *F*. Whichever is the field *F*, the roots of unity in *F* are either complex numbers, if the [characteristic](/source/Characteristic_(algebra)) of *F* is 0, or, otherwise, belong to a [finite field](/source/Finite_field). Conversely, every nonzero element in a finite field is a root of unity in that field. See [Root of unity modulo *n*](/source/Root_of_unity_modulo_n) and [Finite field](/source/Finite_field) for further details.

An nth root of unity is said to be **primitive** if it is not an mth root of unity for some smaller m, that is if[4][5]

- z n = 1 and z m ≠ 1 for m = 1 , 2 , 3 , … , n − 1. {\displaystyle z^{n}=1\quad {\text{and}}\quad z^{m}\neq 1{\text{ for }}m=1,2,3,\ldots ,n-1.}

If *n* is a [prime number](/source/Prime_number), then all *n*th roots of unity, except 1, are primitive.[6]

In the above formula in terms of exponential and trigonometric functions, the primitive nth roots of unity are those for which k and n are [coprime integers](/source/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 [Finite field § Roots of unity](/source/Finite_field#Roots_of_unity). For the case of roots of unity in rings of [modular integers](/source/Modular_arithmetic), see [Root of unity modulo *n*](/source/Root_of_unity_modulo_n).

## Elementary properties

Every *n*th root of unity *z* is a primitive *a*th root of unity for some *a* ≤ *n*, which is the smallest positive integer such that *z**a* = 1.

Any integer power of an *n*th root of unity is also an *n*th root of unity,[7] as

- ( z k ) n = z k n = ( z n ) k = 1 k = 1. {\displaystyle {\bigl (}z^{k}{\bigr )}^{n}=z^{kn}={\bigl (}z^{n}{\bigr )}^{k}=1^{k}=1.}

This is also true for negative exponents. In particular, the [reciprocal](/source/Multiplicative_inverse) of an *n*th root of unity is its [complex conjugate](/source/Complex_conjugate), and is also an *n*th root of unity:[8]

- 1 z = z − 1 = z n − 1 = z ¯ . {\displaystyle {\frac {1}{z}}=z^{-1}=z^{n-1}={\bar {z}}.}

If *z* is an *n*th root of unity and *a* ≡ *b* (mod *n*) then *z**a* = *z**b*. Indeed, by the definition of [congruence modulo *n*](/source/Modular_arithmetic), *a* = *b* + *kn* for some integer *k*, and hence

- z a = z b + k n = z b z k n = z b ( z n ) k = z b 1 k = z b . {\displaystyle z^{a}=z^{b+kn}=z^{b}z^{kn}=z^{b}{\bigl (}z^{n}{\bigr )}^{k}=z^{b}1^{k}=z^{b}.}

Therefore, given a power *z**a* of *z*, one has *z**a* = *z**r*, where 0 ≤ *r* < *n* is the remainder of the [Euclidean division](/source/Euclidean_division) of a by n.

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

From the preceding, it follows that, if *z* is a primitive *n*th root of unity, then z a = z b {\displaystyle z^{a}=z^{b}} [if and only if](/source/If_and_only_if) a ≡ b ( mod n ) . {\displaystyle a\equiv b{\pmod {n}}.} If *z* is not primitive then a ≡ b ( mod n ) {\displaystyle a\equiv b{\pmod {n}}} implies z a = z b , {\displaystyle z^{a}=z^{b},} but the converse may be false, as shown by the following example. If *n* = 4, a non-primitive *n*th root of unity is *z* = −1, and one has z 2 = z 4 = 1 {\displaystyle z^{2}=z^{4}=1} , although 2 ≢ 4 ( mod 4 ) . {\displaystyle 2\not \equiv 4{\pmod {4}}.}

Let *z* be a primitive *n*th root of unity. A power *w* = *z**k* of z is a primitive *a*th root of unity for

- a = n gcd ( k , n ) , {\displaystyle a={\frac {n}{\gcd(k,n)}},}

where gcd ( k , n ) {\displaystyle \gcd(k,n)} is the [greatest common divisor](/source/Greatest_common_divisor) of n and k. This results from the fact that *ka* is the smallest multiple of k that is also a multiple of n. In other words, *ka* is the [least common multiple](/source/Least_common_multiple) of k and n. Thus

- a = lcm ⁡ ( k , n ) k = k n k gcd ( k , n ) = n gcd ( k , n ) . {\displaystyle a={\frac {\operatorname {lcm} (k,n)}{k}}={\frac {kn}{k\gcd(k,n)}}={\frac {n}{\gcd(k,n)}}.}

Thus, if *k* and *n* are [coprime](/source/Coprime), *zk* is also a primitive *n*th root of unity, and therefore there are *φ*(*n*) distinct primitive *n*th roots of unity (where *φ* is [Euler's totient function](/source/Euler's_totient_function)). This implies that if *n* is a prime number, all the roots except +1 are primitive.

In other words, if R(*n*) is the set of all *n*th roots of unity and P(*n*) is the set of primitive ones, R(*n*) is a [disjoint union](/source/Disjoint_union) of the P(*n*):

- R ⁡ ( n ) = ⋃ d | n P ⁡ ( d ) , {\displaystyle \operatorname {R} (n)=\bigcup _{d\,|\,n}\operatorname {P} (d),}

where the notation means that *d* goes through all the positive [divisors](/source/Divisor) of *n*, including 1 and *n*.

Since the [cardinality](/source/Cardinality) of R(*n*) is *n*, and that of P(*n*) is *φ*(*n*), this demonstrates the classical formula

- ∑ d | n φ ( d ) = n . {\displaystyle \sum _{d\,|\,n}\varphi (d)=n.}

## Group properties

### Group of all roots of unity

The product and the [multiplicative inverse](/source/Multiplicative_inverse) of two roots of unity are also roots of unity. In fact, if *xm* = 1 and *yn* = 1, then (*x*−1)*m* = 1, and (*xy*)*k* = 1, where *k* is the [least common multiple](/source/Least_common_multiple) of *m* and *n*.

Therefore, the roots of unity form an [abelian group](/source/Abelian_group) under multiplication. This [group](/source/Group_(mathematics)) is the [torsion subgroup](/source/Torsion_subgroup) of the [circle group](/source/Circle_group).

### Group of *n*th roots of unity

For an integer *n*, the product and the multiplicative inverse of two *n*th roots of unity are also *n*th roots of unity. Therefore, the *n*th roots of unity form an abelian group under multiplication.

Given a primitive *n*th root of unity *ω*, the other *n*th roots are powers of *ω*. This means that the group of the *n*th roots of unity is a [cyclic group](/source/Cyclic_group). It is worth remarking that the term of *cyclic group* originated from the fact that this group is a [subgroup](/source/Subgroup) of the [circle group](/source/Circle_group).

### Galois group of the primitive *n*th roots of unity

Let Q ( ω ) {\displaystyle \mathbb {Q} (\omega )} be the [field extension](/source/Field_extension) of the [rational numbers](/source/Rational_number) generated over Q {\displaystyle \mathbb {Q} } by a primitive *n*th root of unity *ω*. As every *n*th root of unity is a power of *ω*, the [field](/source/Field_(mathematics)) Q ( ω ) {\displaystyle \mathbb {Q} (\omega )} contains all *n*th roots of unity, and Q ( ω ) {\displaystyle \mathbb {Q} (\omega )} is a [Galois extension](/source/Galois_extension) of Q . {\displaystyle \mathbb {Q} .}

If *k* is an integer, *ωk* is a primitive *n*th root of unity if and only if *k* and *n* are [coprime](/source/Coprime). In this case, the map

- ω ↦ ω k {\displaystyle \omega \mapsto \omega ^{k}}

induces an [automorphism](/source/Field_automorphism) of Q ( ω ) {\displaystyle \mathbb {Q} (\omega )} , which maps every *n*th root of unity to its *k*th power. Every automorphism of Q ( ω ) {\displaystyle \mathbb {Q} (\omega )} is obtained in this way, and these automorphisms form the [Galois group](/source/Galois_group) of Q ( ω ) {\displaystyle \mathbb {Q} (\omega )} over the field of the rationals.

The rules of exponentiation imply that the [composition](/source/Function_composition) of two such automorphisms is obtained by multiplying the exponents. It follows that the map

- k ↦ ( ω ↦ ω k ) {\displaystyle k\mapsto \left(\omega \mapsto \omega ^{k}\right)}

defines a [group isomorphism](/source/Group_isomorphism) between the [units](/source/Unit_(ring_theory)) of the ring of [integers modulo *n*](/source/Integers_modulo_n) and the Galois group of Q ( ω ) . {\displaystyle \mathbb {Q} (\omega ).}

This shows that this Galois group is [abelian](/source/Abelian_group), and implies thus that the primitive roots of unity may be expressed in terms of [radicals](/source/Radical_expression).

### Galois group of the real part of the primitive roots of unity

Main article: [Minimal polynomial of 2cos(2pi/n)](/source/Minimal_polynomial_of_2cos(2pi%2Fn))

The real part of the primitive roots of unity are related to one another as roots of the [minimal polynomial](/source/Minimal_polynomial_(field_theory)) of 2 cos ⁡ ( 2 π / n ) . {\displaystyle 2\cos(2\pi /n).} The roots of the minimal polynomial are just twice the real part; these roots form a cyclic Galois group.

## Trigonometric expression

The cube roots of unity

[De Moivre's formula](/source/De_Moivre's_formula), which is valid for all [real](/source/Real_number) x and integers n, is

- ( cos ⁡ x + i sin ⁡ x ) n = cos ⁡ n x + i sin ⁡ n x . {\displaystyle \left(\cos x+i\sin x\right)^{n}=\cos nx+i\sin nx.}

Setting *x* = ⁠2π/*n*⁠ gives a primitive nth root of unity – one gets

- ( cos ⁡ 2 π n + i sin ⁡ 2 π n ) n = cos ⁡ 2 π + i sin ⁡ 2 π = 1 , {\displaystyle \left(\cos {\frac {2\pi }{n}}+i\sin {\frac {2\pi }{n}}\right)^{\!n}=\cos 2\pi +i\sin 2\pi =1,}

but

- ( cos ⁡ 2 π n + i sin ⁡ 2 π n ) k = cos ⁡ 2 k π n + i sin ⁡ 2 k π n ≠ 1 {\displaystyle \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}

for *k* = 1, 2, …, *n* − 1. In other words,

- cos ⁡ 2 π n + i sin ⁡ 2 π n {\displaystyle \cos {\frac {2\pi }{n}}+i\sin {\frac {2\pi }{n}}}

is a primitive nth root of unity.

This formula shows that in the [complex plane](/source/Complex_plane) the nth roots of unity are at the vertices of a [regular n-sided polygon](/source/Regular_polygon) inscribed in the [unit circle](/source/Unit_circle), with one vertex at 1 (see the plot for *n* = 3 on the right). This geometric fact accounts for the term "cyclotomic" in such phrases as [cyclotomic field](/source/Cyclotomic_field) and [cyclotomic polynomial](/source/Cyclotomic_polynomial); it is from the Greek roots "[cyclo](https://en.wiktionary.org/wiki/%CE%BA%CF%8D%CE%BA%CE%BB%CE%BF%CF%82)" (circle) plus "[tomos](https://en.wiktionary.org/wiki/%CF%84%CF%8C%CE%BC%CE%BF%CF%82)" (cut, divide).

[Euler's formula](/source/Euler's_formula)

- e i x = cos ⁡ x + i sin ⁡ x , {\displaystyle e^{ix}=\cos x+i\sin x,}

which is valid for all real x, can be used to put the formula for the nth roots of unity into the form

- e 2 π i k n , 0 ≤ k < n . {\displaystyle e^{2\pi i{\frac {k}{n}}},\quad 0\leq k<n.}

It follows from the discussion in the previous section that this is a primitive nth-root if and only if the fraction ⁠*k*/*n*⁠ is in lowest terms; that is, that k and n are coprime. An [irrational number](/source/Irrational_number) that can be expressed as the [real part](/source/Complex_number) of the root of unity; that is, as cos ⁡ ( 2 π k / n ) {\displaystyle \cos(2\pi k/n)} , is called a [trigonometric number](/source/Trigonometric_number).

## Algebraic expression

The *n*th roots of unity are, by definition, the [roots](/source/Root_of_a_polynomial) of the [polynomial](/source/Polynomial) *xn* − 1, and are thus [algebraic numbers](/source/Algebraic_number). As this polynomial is not [irreducible](/source/Irreducible_polynomial) (except for *n* = 1), the primitive *n*th roots of unity are roots of an irreducible polynomial (over the integers) of lower degree, called the *n*th [cyclotomic polynomial](/source/Cyclotomic_polynomial), and often denoted Φ*n*. The degree of Φ*n* is given by [Euler's totient function](/source/Euler's_totient_function), which counts (among other things) the number of primitive *n*th roots of unity.[9] The roots of Φ*n* are exactly the primitive *n*th roots of unity.

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

Gauss [proved](/source/Mathematical_proof) that a primitive nth root of unity can be expressed using only [square roots](/source/Square_root), addition, subtraction, multiplication and division if and only if it is possible to [construct with compass and straightedge](/source/Compass-and-straightedge_construction) the [regular n-gon](/source/Regular_polygon). This is the case [if and only if](/source/If_and_only_if) *n* is either a [power of two](/source/Power_of_two) or the product of a power of two and [Fermat primes](/source/Fermat_prime) that are all different.

If z is a primitive nth root of unity, the same is true for 1/*z*, and r = z + 1 z {\displaystyle r=z+{\frac {1}{z}}} is twice the real part of z. In other words, Φ*n* is a [reciprocal polynomial](/source/Reciprocal_polynomial), the polynomial R n {\displaystyle R_{n}} that has r as a root may be deduced from Φ*n* by the standard manipulation on reciprocal polynomials, and the primitive nth roots of unity may be deduced from the roots of R n {\displaystyle R_{n}} by solving the [quadratic equation](/source/Quadratic_equation) z 2 − r z + 1 = 0. {\displaystyle z^{2}-rz+1=0.} That is, the real part of the primitive root is r 2 , {\displaystyle {\frac {r}{2}},} and its imaginary part is ± i 1 − ( r 2 ) 2 . {\displaystyle \pm i{\sqrt {1-\left({\frac {r}{2}}\right)^{2}}}.}

The polynomial R n {\displaystyle R_{n}} is an irreducible polynomial whose roots are all real. Its degree is a power of two, if and only if n is a product of a power of two by a product (possibly [empty](/source/Empty_product)) of distinct Fermat primes, and the regular n-gon is constructible with compass and straightedge. Otherwise, it is solvable in radicals, but one are in the [casus irreducibilis](/source/Casus_irreducibilis), that is, every expression of the roots in terms of radicals involves *nonreal radicals*.

### Explicit expressions in low degrees

- For *n* = 1, the cyclotomic polynomial is Φ1(*x*) = *x* − 1 Therefore, the only primitive first root of unity is 1, which is a non-primitive *n*th root of unity for every *n* > 1.

- As Φ2(*x*) = *x* + 1, the only primitive second (square) root of unity is −1, which is also a non-primitive *n*th root of unity for every even *n* > 2. With the preceding case, this completes the list of [real](/source/Real_number) roots of unity.

- As Φ3(*x*) = *x*2 + *x* + 1, the primitive third ([cube](/source/Cube_root)) roots of unity, which are the roots of this [quadratic polynomial](/source/Quadratic_polynomial), are − 1 + i 3 2 , − 1 − i 3 2 . {\displaystyle {\frac {-1+i{\sqrt {3}}}{2}},\ {\frac {-1-i{\sqrt {3}}}{2}}.}

- As Φ4(*x*) = *x*2 + 1, the two primitive fourth roots of unity are *i* and −*i*.

- As Φ5(*x*) = *x*4 + *x*3 + *x*2 + *x* + 1, the four primitive fifth roots of unity are the roots of this [quartic polynomial](/source/Quartic_polynomial), which may be explicitly solved in terms of radicals, giving the roots ε 5 − 1 4 ± i 10 + 2 ε 5 4 , {\displaystyle {\frac {\varepsilon {\sqrt {5}}-1}{4}}\pm i{\frac {\sqrt {10+2\varepsilon {\sqrt {5}}}}{4}},} where ε {\displaystyle \varepsilon } may take the two values 1 and −1 (the same value in the two occurrences).

- As Φ6(*x*) = *x*2 − *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: 1 + i 3 2 , 1 − i 3 2 . {\displaystyle {\frac {1+i{\sqrt {3}}}{2}},\ {\frac {1-i{\sqrt {3}}}{2}}.}

- As 7 is not a Fermat prime, the seventh roots of unity are the first that require [cube roots](/source/Cube_root). There are 6 primitive seventh roots of unity, which are pairwise [complex conjugate](/source/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 r 3 + r 2 − 2 r − 1 , {\displaystyle r^{3}+r^{2}-2r-1,} and the primitive seventh roots of unity are r 2 ± i 1 − r 2 4 , {\displaystyle {\frac {r}{2}}\pm i{\sqrt {1-{\frac {r^{2}}{4}}}},} where 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](/source/Casus_irreducibilis), and any such expression involves non-real cube roots.

- As Φ8(*x*) = *x*4 + 1, the four primitive eighth roots of unity are the square roots of the primitive fourth roots, ± *i*. They are thus ± 2 2 ± i 2 2 . {\displaystyle \pm {\frac {\sqrt {2}}{2}}\pm i{\frac {\sqrt {2}}{2}}.}

- See [Heptadecagon](/source/Heptadecagon) for the real part of a 17th root of unity.

## Periodicity

If z is a primitive nth root of unity, then the sequence of powers

- … , *z*−1, *z*0, *z*1, …

is n-periodic (because *z**j* + *n* = *z**j**z**n* = *z**j* for all values of j), and the n sequences of powers

- *sk*: … , *z**k*⋅(−1), *z**k*⋅0, *z**k*⋅1, …

for *k* = 1, … , *n* are all n-periodic (because *z**k*⋅(*j* + *n*) = *z**k*⋅*j*). Furthermore, the set {*s*1, … , *sn*} of these sequences is a [basis](/source/Basis_(linear_algebra)) of the [linear space](/source/Linear_space) of all n-periodic sequences. This means that *any* n-periodic sequence of complex numbers

- … , *x*−1 , *x*0 , *x*1, …

can be expressed as a [linear combination](/source/Linear_combination) of powers of a primitive nth root of unity:

- x j = ∑ k X k ⋅ z k ⋅ j = X 1 z 1 ⋅ j + ⋯ + X n ⋅ z n ⋅ j {\displaystyle 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}}

for some complex numbers *X*1, … , *X**n* and every integer j.

This is a form of [Fourier analysis](/source/Fourier_analysis). If j is a (discrete) time variable, then k is a [frequency](/source/Frequency) and *X**k* is a complex [amplitude](/source/Amplitude).

Choosing for the primitive nth root of unity

- z = e 2 π i n = cos ⁡ 2 π n + i sin ⁡ 2 π n {\displaystyle z=e^{\frac {2\pi i}{n}}=\cos {\frac {2\pi }{n}}+i\sin {\frac {2\pi }{n}}}

allows *x**j* to be expressed as a linear combination of cos and sin:

- x j = ∑ k A k cos ⁡ 2 π j k n + ∑ k B k sin ⁡ 2 π j k n . {\displaystyle x_{j}=\sum _{k}A_{k}\cos {\frac {2\pi jk}{n}}+\sum _{k}B_{k}\sin {\frac {2\pi jk}{n}}.}

This is a [discrete Fourier transform](/source/Discrete_Fourier_transform).

## Summation

Let SR(*n*) be the sum of all the nth roots of unity, primitive or not. Then

- SR ⁡ ( n ) = { 1 , n = 1 0 , n > 1. {\displaystyle \operatorname {SR} (n)={\begin{cases}1,&n=1\\0,&n>1.\end{cases}}}

This is an immediate consequence of [Vieta's formulas](/source/Vieta's_formulas). In fact, the nth roots of unity being the roots of the polynomial *X* *n* − 1, their sum is the [coefficient](/source/Coefficient) of degree *n* − 1, which is either 1 or 0 according whether *n* = 1 or *n* > 1.

Alternatively, for *n* = 1 there is nothing to prove, and for *n* > 1 there exists a root *z* ≠ 1 – since the set *S* of all the nth roots of unity is a [group](/source/Group_(mathematics)), *z S* = *S*, so the sum satisfies *z* SR(*n*) = SR(*n*), whence SR(*n*) = 0.

Let SP(*n*) be the sum of all the primitive nth roots of unity. Then

- SP ⁡ ( n ) = μ ( n ) , {\displaystyle \operatorname {SP} (n)=\mu (n),}

where *μ*(*n*) is the [Möbius function](/source/M%C3%B6bius_function).

In the section [Elementary properties](#Elementary_properties), it was shown that if R(*n*) is the set of all nth roots of unity and P(*n*) is the set of primitive ones, R(*n*) is a disjoint union of the P(*n*):

- R ⁡ ( n ) = ⋃ d | n P ⁡ ( d ) , {\displaystyle \operatorname {R} (n)=\bigcup _{d\,|\,n}\operatorname {P} (d),}

This implies

- SR ⁡ ( n ) = ∑ d | n SP ⁡ ( d ) . {\displaystyle \operatorname {SR} (n)=\sum _{d\,|\,n}\operatorname {SP} (d).}

Applying the [Möbius inversion formula](/source/M%C3%B6bius_inversion_formula) gives

- SP ⁡ ( n ) = ∑ d | n μ ( d ) SR ⁡ ( n d ) . {\displaystyle \operatorname {SP} (n)=\sum _{d\,|\,n}\mu (d)\operatorname {SR} \left({\frac {n}{d}}\right).}

In this formula, if *d* < *n*, then SR(⁠*n*/*d*⁠) = 0, and for *d* = *n*: SR(⁠*n*/*d*⁠) = 1. Therefore, SP(*n*) = *μ*(*n*).

This is the special case *c**n*(1) of [Ramanujan's sum](/source/Ramanujan's_sum) *c**n*(*s*),[10] defined as the sum of the sth powers of the primitive nth roots of unity:

- c n ( s ) = ∑ a = 1 gcd ( a , n ) = 1 n e 2 π i a n s . {\displaystyle c_{n}(s)=\sum _{a=1 \atop \gcd(a,n)=1}^{n}e^{2\pi i{\frac {a}{n}}s}.}

## Orthogonality

From the summation formula follows an [orthogonality](/source/Orthogonality) relationship: for *j* = 1, … , *n* and *j′* = 1, … , *n*

- ∑ k = 1 n z j ⋅ k ¯ ⋅ z j ′ ⋅ k = n ⋅ δ j , j ′ {\displaystyle \sum _{k=1}^{n}{\overline {z^{j\cdot k}}}\cdot z^{j'\cdot k}=n\cdot \delta _{j,j'}}

where δ is the [Kronecker delta](/source/Kronecker_delta) and z is any primitive nth root of unity.

The *n* × *n* [matrix](/source/Matrix_(mathematics)) U whose (*j*, *k*)th entry is

- U j , k = n − 1 2 ⋅ z j ⋅ k {\displaystyle U_{j,k}=n^{-{\frac {1}{2}}}\cdot z^{j\cdot k}}

defines a [discrete Fourier transform](/source/Discrete_Fourier_transform). Computing the inverse transformation using [Gaussian elimination](/source/Gaussian_elimination) requires *[O](/source/Big-O_notation)*(*n*3) operations. However, it follows from the orthogonality that U is [unitary](/source/Unitary_matrix). That is,

- ∑ k = 1 n U j , k ¯ ⋅ U k , j ′ = δ j , j ′ , {\displaystyle \sum _{k=1}^{n}{\overline {U_{j,k}}}\cdot U_{k,j'}=\delta _{j,j'},}

and thus the inverse of U is simply the complex conjugate. (This fact was first noted by [Gauss](/source/Carl_Friedrich_Gauss) when solving the problem of [trigonometric interpolation](/source/Trigonometric_interpolation).) The straightforward application of U or its inverse to a given vector requires *O*(*n*2) operations. The [fast Fourier transform](/source/Fast_Fourier_transform) algorithms reduces the number of operations further to *O*(*n* log *n*).

## Cyclotomic polynomials

Main article: [Cyclotomic polynomial](/source/Cyclotomic_polynomial)

The [zeros](/source/Zero_of_a_function) of the polynomial

- p ( z ) = z n − 1 {\displaystyle p(z)=z^{n}-1}

are precisely the nth roots of unity, each with [multiplicity](/source/Multiplicity_of_a_root) 1. The nth *[cyclotomic polynomial](/source/Cyclotomic_polynomial)* is defined by the fact that its zeros are precisely the *primitive* nth roots of unity, each with multiplicity 1.

- Φ n ( z ) = ∏ k = 1 φ ( n ) ( z − z k ) {\displaystyle \Phi _{n}(z)=\prod _{k=1}^{\varphi (n)}(z-z_{k})}

where *z*1, *z*2, *z*3, …, *z*φ(*n*) are the primitive nth roots of unity, and φ(*n*) is [Euler's totient function](/source/Euler's_totient_function). The polynomial Φ*n*(*z*) has integer coefficients and is an [irreducible polynomial](/source/Irreducible_polynomial) over the rational numbers (that is, it cannot be written as the product of two positive-degree polynomials with rational coefficients).[9] The case of prime n, which is easier than the general assertion, follows by applying [Eisenstein's criterion](/source/Eisenstein's_criterion) to the polynomial

- ( z + 1 ) n − 1 ( z + 1 ) − 1 , {\displaystyle {\frac {(z+1)^{n}-1}{(z+1)-1}},}

and expanding via the [binomial theorem](/source/Binomial_theorem).

Every nth root of unity is a primitive dth root of unity for exactly one positive [divisor](/source/Divisor) d of n. This implies that[9]

- z n − 1 = ∏ d | n Φ d ( z ) . {\displaystyle z^{n}-1=\prod _{d\,|\,n}\Phi _{d}(z).}

This formula represents the [factorization](/source/Factorization_of_polynomials) of the polynomial *z**n* − 1 into irreducible factors:

- 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 ) {\displaystyle {\begin{aligned}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{aligned}}}

Applying [Möbius inversion](/source/M%C3%B6bius_inversion) to the formula gives

- Φ n ( z ) = ∏ d | n ( z n d − 1 ) μ ( d ) = ∏ d | n ( z d − 1 ) μ ( n d ) , {\displaystyle \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)},}

where *μ* is the [Möbius function](/source/M%C3%B6bius_function). So the first few cyclotomic polynomials are

- Φ1(*z*) = *z* − 1

- Φ2(*z*) = (*z*2 − 1)⋅(*z* − 1)−1 = *z* + 1

- Φ3(*z*) = (*z*3 − 1)⋅(*z* − 1)−1 = *z*2 + *z* + 1

- Φ4(*z*) = (*z*4 − 1)⋅(*z*2 − 1)−1 = *z*2 + 1

- Φ5(*z*) = (*z*5 − 1)⋅(*z* − 1)−1 = *z*4 + *z*3 + *z*2 + *z* + 1

- Φ6(*z*) = (*z*6 − 1)⋅(*z*3 − 1)−1⋅(*z*2 − 1)−1⋅(*z* − 1) = *z*2 − *z* + 1

- Φ7(*z*) = (*z*7 − 1)⋅(*z* − 1)−1 = *z*6 + *z*5 + *z*4 + *z*3 + *z*2 +*z* + 1

- Φ8(*z*) = (*z*8 − 1)⋅(*z*4 − 1)−1 = *z*4 + 1

If p is a [prime number](/source/Prime_number), then all the pth roots of unity except 1 are primitive pth roots. Therefore,[6] Φ p ( z ) = z p − 1 z − 1 = ∑ k = 0 p − 1 z k . {\displaystyle \Phi _{p}(z)={\frac {z^{p}-1}{z-1}}=\sum _{k=0}^{p-1}z^{k}.} Substituting any positive integer ≥ 2 for z, this sum becomes a [base z](/source/Radix) [repunit](/source/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 Φ[105](/source/105_(number)). It is not a surprise it takes this long to get an example, because the behavior of the coefficients depends not so much on n as on how many [odd](/source/Parity_(mathematics)) prime factors appear in n. More precisely, it can be shown that if n has 1 or 2 odd prime factors (for example, *n* = 150) then the nth cyclotomic polynomial only has coefficients 0, 1 or −1. Thus the first conceivable 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 3 ⋅ 5 ⋅ 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](/source/Absolute_value). In particular, if n = p 1 p 2 ⋯ p t , {\displaystyle n=p_{1}p_{2}\cdots p_{t},} where p 1 < p 2 < ⋯ < p t {\displaystyle p_{1}<p_{2}<\cdots <p_{t}} are odd primes, p 1 + p 2 > p t , {\displaystyle p_{1}+p_{2}>p_{t},} and *t* is odd, then 1 − *t* occurs as a coefficient in the nth cyclotomic polynomial.[11]

Many restrictions are known about the values that cyclotomic polynomials can assume at integer values. For example, if p is prime, then *d* ∣ Φ*p*(*d*) if and only if *d* ≡ 1 (mod *p*).

Cyclotomic polynomials are solvable in [radicals](/source/Radical_expression), as roots of unity are themselves radicals. Moreover, there exist more informative radical expressions for nth roots of unity with the additional property[12] that every value of the expression obtained by choosing values of the radicals (for example, signs of square roots) is a primitive nth root of unity. This was already shown by [Gauss](/source/Carl_Friedrich_Gauss) in 1797.[13] Efficient [algorithms](/source/Algorithm) exist for calculating such expressions.[14]

## Cyclic groups

The nth roots of unity form under multiplication a [cyclic group](/source/Cyclic_group) of [order](/source/Order_(group_theory)) n, and in fact these groups comprise all of the [finite](/source/Finite_group) subgroups of the [multiplicative group](/source/Multiplicative_group) of the complex number field. A [generator](/source/Generating_set_of_a_group) for this cyclic group is a primitive nth root of unity.

The nth roots of unity form an irreducible [representation](/source/Group_representation) of any cyclic group of order n. The orthogonality relationship also follows from [group-theoretic](/source/Group_theory) principles as described in [Character group](/source/Character_group).

The roots of unity appear as entries of the [eigenvectors](/source/Eigenvector) of any [circulant matrix](/source/Circulant_matrix); that is, matrices that are invariant under cyclic shifts, a fact that also follows from [group representation theory](/source/Group_representation_theory) as a variant of [Bloch's theorem](/source/Bloch's_theorem).[15][*[page needed](https://en.wikipedia.org/wiki/Wikipedia:Citing_sources)*] In particular, if a circulant [Hermitian matrix](/source/Hermitian_matrix) is considered (for example, a discretized one-dimensional [Laplacian](/source/Laplacian) with periodic boundaries[16]), the orthogonality property immediately follows from the usual orthogonality of eigenvectors of Hermitian matrices.

## Cyclotomic fields

Main article: [Cyclotomic field](/source/Cyclotomic_field)

By [adjoining](/source/Field_extension#adjunction) a primitive nth root of unity to Q , {\displaystyle \mathbb {Q} ,} one obtains the nth [cyclotomic field](/source/Cyclotomic_field) Q ( exp ⁡ ( 2 π i / n ) ) . {\displaystyle \mathbb {Q} (\exp(2\pi i/n)).} This [field](/source/Field_(mathematics)) contains all nth roots of unity and is the [splitting field](/source/Splitting_field) of the nth cyclotomic polynomial over Q . {\displaystyle \mathbb {Q} .} The [field extension](/source/Field_extension) Q ( exp ⁡ ( 2 π i / n ) ) / Q {\displaystyle \mathbb {Q} (\exp(2\pi i/n))/\mathbb {Q} } has degree φ(*n*) and its [Galois group](/source/Galois_group) is [naturally](/source/Natural_transformation) [isomorphic](/source/Group_isomorphism) to the multiplicative [group of units](/source/Group_of_units) of the ring Z / n Z . {\displaystyle \mathbb {Z} /n\mathbb {Z} .}

As the Galois group of Q ( exp ⁡ ( 2 π i / n ) ) / Q {\displaystyle \mathbb {Q} (\exp(2\pi i/n))/\mathbb {Q} } is abelian, this is an [abelian extension](/source/Abelian_extension). Every [subfield](/source/Field_extension) 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](/source/Galois_theory) can be written out explicitly in terms of [Gaussian periods](/source/Gaussian_period): this theory from the *[Disquisitiones Arithmeticae](/source/Disquisitiones_Arithmeticae)* of [Gauss](/source/Carl_Friedrich_Gauss) was published many years before Galois.[17]

Conversely, *every* abelian extension of the rationals is such a subfield of a cyclotomic field – this is the content of a theorem of [Kronecker](/source/Leopold_Kronecker), usually called the *[Kronecker–Weber theorem](/source/Kronecker%E2%80%93Weber_theorem)* on the grounds that Weber completed the proof.

## Relation to quadratic integers

In the [complex plane](/source/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.

In the complex plane, the corners of the two squares are the eighth roots of unity.

For *n* = 1, 2, both roots of unity [1](/source/1_(number)) and [−1](/source/%E2%88%921_(number)) are [integers](/source/Integer).

For three values of n, the roots of unity are [quadratic integers](/source/Quadratic_integer):

- For *n* = 3, 6 they are [Eisenstein integers](/source/Eisenstein_integer) ([*D*](/source/Discriminant) = −3).

- For *n* = 4 they are [Gaussian integers](/source/Gaussian_integer) (*D* = −1): see [Imaginary unit](/source/Imaginary_unit).

For four other values of n, the primitive roots of unity are not quadratic integers, but the sum of any root of unity with its [complex conjugate](/source/Complex_conjugate) (also an nth root of unity) is a quadratic integer.

For *n* = 5, 10, none of the non-real roots of unity (which satisfy a [quartic equation](/source/Quartic_equation)) is a quadratic integer, but the sum *z* + *z* = 2 [Re](/source/Real_part) *z* of each root with its complex conjugate (also a 5th root of unity) is an element of the [ring](/source/Ring_(mathematics)) [**Z**\[⁠1 + √5/2⁠\]](/source/Quadratic_integer) (*D* = 5). For two pairs of non-real 5th roots of unity these sums are [inverse](/source/Multiplicative_inverse) [golden ratio](/source/Golden_ratio) and [minus](/source/Additive_inverse) golden ratio.

For *n* = 8, for any root of unity *z* + *z* equals to either 0, ±2, or ±[√2](/source/Square_root_of_2) (*D* = 2).

For *n* = 12, for any root of unity, *z* + *z* equals to either 0, ±1, ±2 or ±[√3](/source/Square_root_of_3) (*D* = 3).

## See also

- [Argand system](/source/Argand_system)

- [Circle group](/source/Circle_group), the unit complex numbers

- [Cyclotomic field](/source/Cyclotomic_field)

- [Group scheme of roots of unity](/source/Group_scheme_of_roots_of_unity)

- [Dirichlet character](/source/Dirichlet_character)

- [Ramanujan's sum](/source/Ramanujan's_sum)

- [Witt vector](/source/Witt_vector)

- [Teichmüller character](/source/Teichm%C3%BCller_character)

## Notes

1. **[^](#cite_ref-1)** [Hadlock, Charles R.](/source/Charles_Robert_Hadlock) (2000). [*Field Theory and Its Classical Problems, Volume 14*](https://books.google.com/books?id=5s1p0CyafnEC&pg=PA84). Cambridge University Press. pp. 84–86. [ISBN](/source/ISBN_(identifier)) [978-0-88385-032-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-88385-032-9).

1. **[^](#cite_ref-2)** Lang, Serge (2002). ["Roots of unity"](https://books.google.com/books?id=Fge-BwqhqIYC&pg=PA276). *Algebra*. Springer. pp. 276–277. [ISBN](/source/ISBN_(identifier)) [978-0-387-95385-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-95385-4).

1. **[^](#cite_ref-meserve_3-0)** Meserve, Bruce E. (1982). *Fundamental Concepts of Algebra*. Dover Publications. p. 52.

1. **[^](#cite_ref-moskowitz_4-0)** Moskowitz, Martin A. (2003). [*Adventure in Mathematics*](https://books.google.com/books?id=YT2_Kqsnn9wC&pg=PA36). World Scientific. p. 36. [ISBN](/source/ISBN_(identifier)) [9789812794949](https://en.wikipedia.org/wiki/Special:BookSources/9789812794949).

1. **[^](#cite_ref-lidl_5-0)** Lidl, Rudolf; [Pilz, Günter](/source/G%C3%BCnter_Pilz) (1984). [*Applied Abstract Algebra*](https://books.google.com/books?id=irXSBwAAQBAJ&pg=PA149). Undergraduate Texts in Mathematics. Springer. p. 149. [doi](/source/Doi_(identifier)):[10.1007/978-1-4615-6465-2](https://doi.org/10.1007%2F978-1-4615-6465-2). [ISBN](/source/ISBN_(identifier)) [978-0-387-96166-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-96166-8).

1. ^ [***a***](#cite_ref-morandi_6-0) [***b***](#cite_ref-morandi_6-1) Morandi, Patrick (1996). [*Field and Galois theory*](https://books.google.com/books?id=jQ7c8Xqpqk0C&pg=PA74). Graduate Texts in Mathematics. Vol. 167. Springer. p. 74. [doi](/source/Doi_(identifier)):[10.1007/978-1-4612-4040-2](https://doi.org/10.1007%2F978-1-4612-4040-2). [ISBN](/source/ISBN_(identifier)) [978-0-387-94753-2](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-94753-2).

1. **[^](#cite_ref-integer-power_7-0)** Reilly, Norman R. (2009). [*Introduction to Applied Algebraic Systems*](https://books.google.com/books?id=q33he4hOlKcC&pg=PA137). Oxford University Press. p. 137. [ISBN](/source/ISBN_(identifier)) [978-0-19-536787-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-19-536787-4).

1. **[^](#cite_ref-conjugate_8-0)** [Rotman, Joseph J.](/source/Joseph_J._Rotman) (2015). [*Advanced Modern Algebra*](https://books.google.com/books?id=SugUCwAAQBAJ&pg=PA129). Vol. 1 (3rd ed.). American Mathematical Society. p. 129. [ISBN](/source/ISBN_(identifier)) [9781470415549](https://en.wikipedia.org/wiki/Special:BookSources/9781470415549).

1. ^ [***a***](#cite_ref-riesel_9-0) [***b***](#cite_ref-riesel_9-1) [***c***](#cite_ref-riesel_9-2) [Riesel, Hans](/source/Hans_Riesel) (1994). [*Prime Factorization and Computer Methods for Factorization*](https://books.google.com/books?id=ITvaBwAAQBAJ&pg=PA306). Springer. p. 306. [ISBN](/source/ISBN_(identifier)) [0-8176-3743-5](https://en.wikipedia.org/wiki/Special:BookSources/0-8176-3743-5).

1. **[^](#cite_ref-apostol_10-0)** [Apostol, Tom M.](/source/Tom_M._Apostol) (1976). [*Introduction to Analytic Number Theory*](https://books.google.com/books?id=3yoBCAAAQBAJ&pg=PA160). Undergraduate Texts in Mathematics. Springer. p. 160. [doi](/source/Doi_(identifier)):[10.1007/978-1-4757-5579-4](https://doi.org/10.1007%2F978-1-4757-5579-4). [ISBN](/source/ISBN_(identifier)) [978-1-4419-2805-4](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4419-2805-4).

1. **[^](#cite_ref-lehmer_11-0)** [Lehmer, Emma](/source/Emma_Lehmer) (1936). ["On the magnitude of the coefficients of the cyclotomic polynomial"](https://doi.org/10.1090%2FS0002-9904-1936-06309-3). *Bulletin of the American Mathematical Society*. **42** (6): 389–392. [doi](/source/Doi_(identifier)):[10.1090/S0002-9904-1936-06309-3](https://doi.org/10.1090%2FS0002-9904-1936-06309-3).

1. **[^](#cite_ref-landau_12-0)** [Landau, Susan](/source/Susan_Landau); [Miller, Gary L.](/source/Gary_Miller_(computer_scientist)) (1985). "Solvability by radicals is in polynomial time". *Journal of Computer and System Sciences*. **30** (2): 179–208. [doi](/source/Doi_(identifier)):[10.1016/0022-0000(85)90013-3](https://doi.org/10.1016%2F0022-0000%2885%2990013-3).

1. **[^](#cite_ref-13)** [Gauss, Carl F.](/source/Carl_Friedrich_Gauss) (1965). *Disquisitiones Arithmeticae*. Yale University Press. pp. §§359–360. [ISBN](/source/ISBN_(identifier)) [0-300-09473-6](https://en.wikipedia.org/wiki/Special:BookSources/0-300-09473-6).

1. **[^](#cite_ref-14)** Weber, Andreas; Keckeisen, Michael. ["Solving Cyclotomic Polynomials by Radical Expressions"](http://cg.cs.uni-bonn.de/personal-pages/weber/publications/pdf/WeberA/WeberKeckeisen99a.pdf) (PDF). Retrieved 22 June 2007.

1. **[^](#cite_ref-yoshitaka_15-0)** Inui, Teturo; Tanabe, Yukito; Onodera, Yoshitaka (1996). *Group Theory and Its Applications in Physics*. Springer.

1. **[^](#cite_ref-siam_16-0)** [Strang, Gilbert](/source/Gilbert_Strang) (1999). ["The discrete cosine transform"](http://epubs.siam.org/sam-bin/dbq/article/33674). *SIAM Review*. **41** (1): 135–147. [Bibcode](/source/Bibcode_(identifier)):[1999SIAMR..41..135S](https://ui.adsabs.harvard.edu/abs/1999SIAMR..41..135S). [doi](/source/Doi_(identifier)):[10.1137/S0036144598336745](https://doi.org/10.1137%2FS0036144598336745).

1. **[^](#cite_ref-17)** The *Disquisitiones* was published in 1801, [Galois](/source/%C3%89variste_Galois) was born in 1811, died in 1832, but wasn't published until 1846.

## References

- [Lang, Serge](/source/Serge_Lang) (2002), *[Algebra](/source/Algebra_(Lang))*, [Graduate Texts in Mathematics](/source/Graduate_Texts_in_Mathematics), vol. 211 (Revised third ed.), New York: Springer-Verlag, [ISBN](/source/ISBN_(identifier)) [978-0-387-95385-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-95385-4), [MR](/source/MR_(identifier)) [1878556](https://mathscinet.ams.org/mathscinet-getitem?mr=1878556), [Zbl](/source/Zbl_(identifier)) [0984.00001](https://zbmath.org/?format=complete&q=an:0984.00001)

- [Milne, James S.](/source/James_Milne_(mathematician)) (1998). ["Algebraic Number Theory"](http://www.jmilne.org/math). *Course Notes*.

- Milne, James S. (1997). ["Class Field Theory"](http://www.jmilne.org/math). *Course Notes*.

- [Neukirch, Jürgen](/source/J%C3%BCrgen_Neukirch) (1999). *Algebraische Zahlentheorie*. *Grundlehren der mathematischen Wissenschaften*. Vol. 322. Berlin: [Springer-Verlag](/source/Springer_Science%2BBusiness_Media). [ISBN](/source/ISBN_(identifier)) [978-3-540-65399-8](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-65399-8). [MR](/source/MR_(identifier)) [1697859](https://mathscinet.ams.org/mathscinet-getitem?mr=1697859). [Zbl](/source/Zbl_(identifier)) [0956.11021](https://zbmath.org/?format=complete&q=an:0956.11021).

- Neukirch, Jürgen (1986). *Class Field Theory*. Berlin: Springer-Verlag. [ISBN](/source/ISBN_(identifier)) [3-540-15251-2](https://en.wikipedia.org/wiki/Special:BookSources/3-540-15251-2).

- [Washington, Lawrence C.](/source/Lawrence_C._Washington) (1997). *Introduction to Cyclotomic Fields* (2nd ed.). New York: Springer-Verlag. [ISBN](/source/ISBN_(identifier)) [0-387-94762-0](https://en.wikipedia.org/wiki/Special:BookSources/0-387-94762-0).

- [Derbyshire, John](/source/John_Derbyshire) (2006). "Roots of Unity". [*Unknown Quantity*](https://archive.org/details/isbn_9780309096577). Washington, D.C.: [Joseph Henry Press](/source/Joseph_Henry_Press). [ISBN](/source/ISBN_(identifier)) [0-309-09657-X](https://en.wikipedia.org/wiki/Special:BookSources/0-309-09657-X).

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