Root of unity

The 5th roots of unity in the complex plane

In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that gives 1 when raised to some positive integer power $n$ . Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform.

In field theory and ring theory the notion of root of unity also applies to any ring with a multiplicative identity element. Any algebraically closed field has exactly $n$ $n$ th roots of unity, if $n$ is not divisible by the characteristic of the field.

General definition

An $n$ th root of unity, where $n$ is a positive integer (i.e. $n = 1, 2, 3, \dots$ ), is a number $z$ satisfying the equation^[1]^[2]

z^n = 1.

Traditionally, $z$ is assumed to be a complex number, and subsequent sections of this article will comply with this usage. Generally, $z \in R$ can be considered for any field $R$ , or even for a unital ring. In this general formulation, an $n$ th root of unity is just an element of the group of units of order $n$ . Interesting cases are finite fields and modular arithmetics, for which the article root of unity modulo n contains some information.

An $n$ th root of unity is primitive if it is not a $k$ th root of unity for some smaller $k$ :

z^k \ne 1 \qquad (k = 1, 2, 3, \dots, n-1 ).

Properties

Every $n$ th root of unity $z$ is a primitive $a$ th root of unity for some $a$ where $1 \leq a \leq n$ : if $z 1 = 1$ then $z$ is a primitive first root of unity, otherwise if $z 2 = 1$ then $z$ is a primitive second (square) root of unity, otherwise, ..., and by assumption there must be a "1" at or before the $n$ th term in the sequence.

If $z$ is an $n$ th root of unity and $a \equiv b (mod n)$ then $z a = z b$ . By the definition of congruence, $a = b + kn$ for some integer $k$ . But then,

z^a = z^{b+kn} = z^b z^{kn} = z^b (z^n)^k = z^b 1^k = z^b.

Therefore, given a power $z a$ of $z$ , it can be assumed that $1 \leq a \leq n$ . This is often convenient.

Any integer power of an $n$ th root of unity is also an $n$ th root of unity:

(z^k)^n = z^{kn} = (z^n)^k = 1^k = 1.

Here $k$ may be negative. In particular, the reciprocal of an $n$ th root of unity is its complex conjugate, and is also an $n$ th root of unity:

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

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 all distinct. Assume the contrary, that $z a = z b$ where $1 \leq a < b \leq n$ . Then $z b - a = 1$ . But $0 < b - a < n$ , which contradicts $z$ being primitive.

Since an $n$ th-degree polynomial equation can only have $n$ distinct roots, this implies that the powers of a primitive root $z$ , $z 2$ , … , $z n - 1$ , $z n = z 0 = 1$ are all of the $n$ th roots of unity.

From the preceding, it follows that if $z$ is a primitive $n$ th root of unity:

z^a = z^b \iff a\equiv b \pmod{ n}.

If $z$ is not primitive there is only one implication:

a\equiv b \pmod{ n} \implies z^a = z^b.

An example showing that the converse implication is false is given by:

n = 4, \;\; z = -1, \;\; z^2 = z^4 = 1, \;\; 2 \not\equiv 4 \pmod{4}.

Let $z$ be a primitive $n$ th root of unity and let $k$ be a positive integer. From the above discussion, $z k$ is a primitive $a$ th root of unity for some $a$ . Now if $z ka = 1$ , $ka$ must be a multiple of $n$ . The smallest number that is divisible by both $n$ and $k$ is their least common multiple, denoted by $lcm(n, k)$ . It is related to their greatest common divisor, $gcd(n, k)$ , by the formula:

k\,n = \gcd(k,n)\, \operatorname{lcm}(k,n),

i.e.

\operatorname{lcm}(k,n) = k \frac{n}{ \gcd(k,n)\, }.

Therefore, $z k$ is a primitive $a$ th root of unity where

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

Thus, if $k$ and $n$ are coprime, $z k$ is also a primitive $n$ th root of unity, and therefore there are $φ(n)$ (where $φ$ is Euler's totient function) distinct primitive $n$ th roots of unity. (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 of the $P(n)$ :

\operatorname{R}(n) = \bigcup_{d\,|\,n}\operatorname{P}(d),

where the notation means that $d$ goes through all the divisors of $n$ , including 1 and $n$ .

Since the cardinality of $R(n)$ is $n$ , and that of $P(n)$ is $φ(n)$ , this demonstrates the classical formula

\sum_{d\,|\,n}\phi(d) = n.

Examples

The 3rd roots of unity

Plot of

z 3 - 1

, in which a zero is represented by the color black.

Plot of

z 5 - 1

, in which a zero is represented by the color black.

De Moivre's formula, which is valid for all real $x$ and integers $n$ , is

(\cos x + i \sin x)^n = \cos nx + i \sin nx.

Setting $x = 2π/ n$ gives a primitive $n$ th root of unity:

\left(\cos\tfrac{2\pi}{n} + i \sin\tfrac{2\pi}{n}\right)^n = \cos 2\pi + i \sin 2\pi = 1,

but for $k = 1, 2, \dots , n - 1$ ,

\left(\cos\tfrac{2\pi}{n} + i \sin\tfrac{2\pi}{n}\right)^k= \cos\tfrac{2k\pi}{n} + i \sin\tfrac{2k\pi}{n} \neq 1

This formula shows that on the complex plane the $n$ th roots of unity are at the vertices of a regular $n$ -sided polygon inscribed in the unit circle, with one vertex at 1. (See the plots for $n = 3$ and $n = 5$ on the right.) This geometric fact accounts for the term "cyclotomic" in such phrases as cyclotomic field and cyclotomic polynomial; it is from the Greek roots "cyclo" (circle) plus "tomos" (cut, divide).

Euler's formula

e^{i x} = \cos x + i \sin x,

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

e^{2\pi i \frac{k}{n}} \qquad 0 \le k < n.

It follows from the discussion in the previous section that this is a primitive $n$ th-root if and only if the fraction $k / n$ is in lowest terms, i.e. that $k$ and $n$ are coprime.

The roots of unity are trigonometric numbers and are, by definition, the roots of a polynomial equation and are thus algebraic numbers. In fact, Galois theory can be used to show that they may be expressed as expressions involving integers and the operations of addition, subtraction, multiplication, division, and the extraction of roots. (There are more details later in this article at Cyclotomic fields.)

The equation $z 1 = 1$ obviously has only one solution, +1, which is therefore the only primitive first root of unity. It is a nonprimitive 2nd, 3rd, 4th, ... root of unity.

The equation $z 2 = 1$ has two solutions, +1 and −1. +1 is the primitive first root of unity, leaving −1 as the only primitive second (square) root of unity. It is a nonprimitive 4th, 6th, 8th, ...root of unity.

The only real roots of unity are ±1; all the others are non-real complex numbers, as can be seen from de Moivre's formula or the figures.

The third (cube) roots satisfy the equation $z 3 - 1 = 0$ ; the non-principal root +1 may be factored out, giving $(z - 1)(z 2 + z + 1) = 0$ . Therefore, the primitive cube roots of unity are the roots of a quadratic equation. (See Cyclotomic polynomial, below.)

\left\{e^{\frac{2 \pi i}{3}},e^{-\frac{2 \pi i}{3}}\right\}=\left\{ \frac{-1 + i \sqrt{3}}{2}, \frac{-1 - i \sqrt{3}}{2} \right\}

The two primitive fourth roots of unity are the two square roots of the primitive square root of unity, −1

\left\{e^{\frac{2 \pi i}{4}},e^{-\frac{2 \pi i}{4}}\right\}=\left\{\pm\sqrt{-1} \right\}=\left\{+i, -i \right\}.

The four primitive fifth roots of unity are

\left\{\left.e^{\frac{2 \pi i k}{5}}\right| 1 \le k \le 4 \right\}=\left\{\left . \frac{u\sqrt 5-1}4+v\,i\,\sqrt{\frac{5+u\sqrt 5}8}\; \right |u,v \in \{-1,1\}\right\}.

The two primitive sixth roots of unity are the negatives (and also the square roots) of the two primitive cube roots:

\left\{e^{\frac{2 \pi i}{6}},e^{-\frac{2 \pi i}{6}}\right\}=\left\{ \frac{1 + i \sqrt{3}}{2}, \frac{1 - i \sqrt{3}}{2} \right\}.

Gauss observed that if a primitive $n$ th root of unity can be expressed using only square roots, then it is possible to construct the regular $n$ -gon using only ruler and compass, and that if the root of unity requires third or fourth or higher radicals the regular polygon cannot be constructed. The 7th roots of unity are the first that require cube roots. Note that the real part and imaginary part are both real numbers, but complex numbers are buried in the expressions. They cannot be removed. See casus irreducibilis for details.

One of the primitive seventh roots of unity is

e^{\frac{2\pi i}{7}}=\frac{-1 + \sqrt[3]{\frac{7+21\sqrt{-3}}{2}} + \sqrt[3]{\frac{7-21\sqrt{-3}}{2}}}{6} + \frac{i}{2}\sqrt{\frac{7 - \omega^2\sqrt[3]{\frac{7+21\sqrt{-3}}{2}} - \omega\sqrt[3]{\frac{7-21\sqrt{-3}}{2}}}{3}}

where $ω$ and $ω 2$ are the primitive cube roots of unity $exp(2π i /3)$ and $exp(4π i /3)$ .

The four primitive eighth roots of unity are ± the square roots of the primitive fourth roots, $\pm i$ . One of them is:

e^{\frac{2\pi i}{8}} = \sqrt{i}= \frac{\sqrt{2}}{2}+i\frac{\sqrt{2}}{2}.

See heptadecagon for the real part of a 17th root of unity.

Periodicity

If $z$ is a primitive $n$ th root of unity, then the sequence of powers

\dots , z -1, z 0, z 1, \dots

is $n$ -periodic (because $z j + n = z j \cdot z n = z j \cdot1 = z j$ for all values of $j$ ), and the $n$ sequences of powers

s k : \dots , z k \cdot(-1), z k \cdot0, z k \cdot1, \dots

for $k = 1, \dots , n$ are all $n$ -periodic (because $z k \cdot(j + n) = z k \cdot j$ ). Furthermore, the set ${s 1, \dots , s n$ } of these sequences is a basis of the linear space of all $n$ -periodic sequences. This means that any $n$ -periodic sequence of complex numbers

\dots , x -1, x 0, x 1, \dots

can be expressed as a linear combination of powers of a primitive $n$ th root of unity:

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 $X 1, \dots , X n$ and every integer $j$ .

This is a form of Fourier analysis. If $j$ is a (discrete) time variable, then $k$ is a frequency and $X k$ is a complex amplitude.

Choosing for the primitive $n$ th root of unity

z = e 2πi/ n = cos(2π/ n) + i \cdotsin(2π/ n)

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

x j = \sum k A k \cdotcos(2π\cdot j \cdot k / n) + \sum k B k \cdotsin(2π\cdot j \cdot k / n)

This is a discrete Fourier transform.

Summation

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

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

For $n = 1$ there is nothing to prove. For $n > 1$ , it is "intuitively obvious" from the symmetry of the roots in the complex plane. For a rigorous proof, let $z$ be a primitive $n$ th root of unity. Then the set of all roots is given by $z k, k = 0, 1, \dots , n - 1$ , and their sum is given by the formula for a geometric series:

\sum_{k=0}^{n-1} z^k = \frac{z^n - 1}{z - 1} = 0.

Let $SP(n)$ be the sum of all the primitive $n$ th roots of unity. Then

\operatorname{SP}(n) = \mu(n),

where $μ(n)$ is the Möbius function.

In the section Elementary facts, it was shown that 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 of the $P(n)$ :

\operatorname{R}(n) = \bigcup_{d\,|\,n}\operatorname{P}(d),

This implies

\operatorname{SR}(n) = \sum_{d\,|\,n}\operatorname{SP}(d).

Applying the Möbius inversion formula gives

\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 $c n (s)$ , defined as the sum of the $s$ th powers of the primitive $n$ th roots of unity:

c_n(s)= \sum_{a=1\atop \gcd(a,n)=1}^n e^{2 \pi i \tfrac{a}{n} s}.

Orthogonality

From the summation formula follows an orthogonality relationship: for $j = 1, \dots , n$ and $j' = 1, \dots , n$

\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 and $z$ is any primitive $n$ th root of unity.

The $n \times n$ matrix $U$ whose $(j, k)$ th entry is

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

defines a discrete Fourier transform. Computing the inverse transformation using gaussian elimination requires $O (n 3)$ operations. However, it follows from the orthogonality that $U$ is unitary. That is,

\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 when solving the problem of trigonometric interpolation). The straightforward application of $U$ or its inverse to a given vector requires $O (n 2)$ operations. The fast Fourier transform algorithms reduces the number of operations further to $O (n log n)$ .

Cyclotomic polynomials

Main article: Cyclotomic polynomial

The zeroes of the polynomial

p(z) = z^n - 1

are precisely the $n$ th roots of unity, each with multiplicity 1. The $n$ th cyclotomic polynomial is defined by the fact that its zeros are precisely the primitive $n$ th roots of unity, each with multiplicity 1.

\Phi_n(z) = \prod_{k=1}^{\varphi(n)}(z-z_k)

where $z 1, z 2, z 3, \dots , z φ(n)$ are the primitive $n$ th roots of unity, and $φ(n)$ is Euler's totient function. The polynomial $Φ n (z)$ has integer coefficients and is an irreducible polynomial over the rational numbers (i.e., it cannot be written as the product of two positive-degree polynomials with rational coefficients). The case of prime $n$ , which is easier than the general assertion, follows by applying Eisenstein's criterion to the polynomial

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

and expanding via the binomial theorem.

Every $n$ th root of unity is a primitive $d$ th root of unity for exactly one positive divisor $d$ of $n$ . This implies that

z^n - 1 = \prod_{d\,\mid\,n} \Phi_d(z).

This formula represents the factorization of the polynomial $z n - 1$ into irreducible factors.

z 1 - 1 = z - 1

z 2 - 1 = (z - 1)\cdot(z + 1)

z 3 - 1 = (z - 1)\cdot(z 2 + z + 1)

z 4 - 1 = (z - 1)\cdot(z + 1)\cdot(z 2 + 1)

z 5 - 1 = (z - 1)\cdot(z 4 + z 3 + z 2 + z + 1)

z 6 - 1 = (z - 1)\cdot(z + 1)\cdot(z 2 + z + 1)\cdot(z 2 - z + 1)

z 7 - 1 = (z - 1)\cdot(z 6 + z 5 + z 4 + z 3 + z 2 + z + 1)

Applying Möbius inversion to the formula gives

\Phi_n(z) = \prod_{d\,\mid n}(z^{n/d}-1)^{\mu(d)} = \prod_{d\,\mid n}(z^{d}-1)^{\mu(n/d)},

where $μ$ is the Möbius function.

So the first few cyclotomic polynomials are

Φ 1 (z) = z - 1

Φ 2 (z) = (z 2 - 1)\cdot(z - 1) -1 = z + 1

Φ 3 (z) = (z 3 - 1)\cdot(z - 1) -1 = z 2 + z + 1

Φ 4 (z) = (z 4 - 1)\cdot(z 2 - 1) -1 = z 2 + 1

Φ 5 (z) = (z 5 - 1)\cdot(z - 1) -1 = z 4 + z 3 + z 2 + z + 1

Φ 6 (z) = (z 6 - 1)\cdot(z 3 - 1) -1 \cdot(z 2 - 1) -1 \cdot(z - 1) = z 2 - z + 1

Φ 7 (z) = (z 7 - 1)\cdot(z - 1) -1 = z 6 + z 5 + z 4 + z 3 + z 2 + z + 1

If $p$ is a prime number, then all the $p$ th roots of unity except 1 are primitive $p$ th roots, and we have

\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 $z$ 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 $Φ 105$ . 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 prime factors appear in $n$ . More precisely, it can be shown that if $n$ has 1 or 2 odd prime factors (e.g., $n = 150$ ) then the $n$ th 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 $3\cdot5\cdot7 = 105$ . 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 $n = p 1 \cdot p 2 \cdot \dots \cdot p t$ , where $p 1 < p 2 < \dots < p t$ are odd primes, $p 1 + p 2 > p t$ , and t is odd, then $1 - t$ occurs as a coefficient in the $n$ th cyclotomic polynomial.^[3]

Many restrictions are known about the values that cyclotomic polynomials can assume at integer values. For example, if $p$ is prime and $d ∣ Φ p (d)$ , then either $d \equiv 1 (mod p)$ , or $d \equiv 0 (mod p)$ .

Cyclotomic polynomials are solvable in radicals, as roots of unity are themselves radicals. Moreover, there exist more informative radical expressions for $n$ th roots of unity with the additional property^[4] that every value of the expression obtained by choosing values of the radicals (for example, signs of square roots) is a primitive $n$ th root of unity. This was already shown by Gauss in 1797.^[5] Efficient algorithms exist for calculating such expressions.^[6]

Cyclic groups

The $n$ th roots of unity form under multiplication a cyclic group of order $n$ , and in fact these groups comprise all of the finite subgroups of the multiplicative group of the complex number field. A generator for this cyclic group is a primitive $n$ th root of unity.

The $n$ th roots of unity form an irreducible representation of any cyclic group of order $n$ . The orthogonality relationship also follows from group-theoretic principles as described in character group.

The roots of unity appear as entries of the eigenvectors of any circulant matrix, i.e. matrices that are invariant under cyclic shifts, a fact that also follows from group representation theory as a variant of Bloch's theorem.^[7] In particular, if a circulant Hermitian matrix is considered (for example, a discretized one-dimensional Laplacian with periodic boundaries^[8]), the orthogonality property immediately follows from the usual orthogonality of eigenvectors of Hermitian matrices.

Cyclotomic fields

Main article: Cyclotomic field

By adjoining a primitive $n$ th root of unity to Q, one obtains the $n$ th cyclotomic field $Q (exp(2π i / n))$ . This field contains all $n$ th roots of unity and is the splitting field of the $n$ th cyclotomic polynomial over Q. The field extension $Q (exp(2π i / n))/ Q$ has degree φ(n) and its Galois group is naturally isomorphic to the multiplicative group of units of the ring $Z / n Z$ .

As the Galois group of $Q (exp(2π i / n))/ Q$ is abelian, this is an abelian extension. Every subfield of a cyclotomic field is an abelian extension of the rationals. It follows that every nth 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 periods: this theory from the Disquisitiones Arithmeticae of Gauss was published many years before Galois.^[9]

Conversely, every abelian extension of the rationals is such a subfield of a cyclotomic field – this is the content of a theorem of Kronecker, usually called the Kronecker–Weber theorem on the grounds that Weber completed the proof.

Relation to quadratic integers

In the complex plane, the red points are the fifth roots of unity, and the blue points are the sums of a fifth root of unit and its complex conjugate.

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

For $n = 2$ , both roots of unity 1 and −1 belong to $Z$ .

For three values of $n$ , the roots of unity are quadratic integers:

For $n = 3, 6$ they are Eisenstein integers ( $D = -3$ ).
For $n = 4$ they are Gaussian integers ( $D = -1$ ): see 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 (also a $n$ th root of unity) is a quadratic integer.

For $n = 5, 10$ , neither of non-real roots of unity (which satisfy a quartic equation) is a quadratic integer, but the sum $z + z = 2 Re z$ of each root with its complex conjugate (also a 5th root of unity) is an element of the ring Z[1 + √5/2] ( $D = 5$ ). For two pairs of non-real 5th roots of unity these sums are inverse golden ratio and minus golden ratio.

For $n = 8$ , for any root of unity $z + z$ equals to either ±2, 0, or ±√2 ( $D = 2$ ).

For $n = 12$ , for any root of unity, $z + z$ equals to either 0, ±1, ±2 or ±√3 ( $D = 3$ ).

Notes

↑ Hadlock, Charles R. (2000). Field Theory and Its Classical Problems, Volume 14. Cambridge University Press. pp. 84–86. ISBN 978-0-88385-032-9.
↑ Lang, Serge (2002). "Roots of unity". Algebra. Springer. pp. 276–277. ISBN 978-0-387-95385-4.
↑ Emma Lehmer, On the magnitude of the coefficients of the cyclotomic polynomial, Bulletin of the American Mathematical Society 42 (1936), no. 6, pp. 389–392.
↑ Landau, Susan; Miller, Gary L. (1985). "Solvability by radicals is in polynomial time". Journal of Computer and System Sciences 30 (2): 179–208. doi:10.1016/0022-0000(85)90013-3.
↑ Gauss, Carl F. (1965). Disquisitiones Arithmeticae. Yale University Press. pp. §§359–360. ISBN 0-300-09473-6.
↑ Weber, Andreas; Keckeisen, Michael. "Solving Cyclotomic Polynomials by Radical Expressions" (PDF). Retrieved 2007-06-22.
↑ T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and Its Applications in Physics (Springer, 1996).
↑ Gilbert Strang, "The discrete cosine transform," SIAM Review 41 (1), 135–147 (1999).
↑ The Disquisitiones was published in 1801, Galois was born in 1811, died in 1832, but wasn't published until 1846.

References

Lang, Serge (2002), Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, Zbl 0984.00001, MR 1878556
Milne, James S. (1998). "Algebraic Number Theory". Course Notes.
Milne, James S. (1997). "Class Field Theory". Course Notes.
Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, Zbl 0956.11021, MR 1697859
Neukirch, Jürgen (1986). Class Field Theory. Berlin: Springer-Verlag. ISBN 3-540-15251-2.
Washington, Lawrence C. (1997). Cyclotomic fields (2nd ed.). New York: Springer-Verlag. ISBN 0-387-94762-0.
Derbyshire, John (2006). "Roots of Unity". Unknown Quantity. Washington, D.C.: Joseph Henry Press. ISBN 0-309-09657-X.