Root of unity

Template:Short description Template:Use American English Template:Use dmy dates Template:More citations needed

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 raised to some positive integer power Template:Mvar. 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. It is occasionally called a de Moivre number after French mathematician Abraham de Moivre.

Roots of unity can be defined in any field. If the characteristic of the field is zero, the roots are complex numbers that are also algebraic integers. For fields with a positive characteristic, the roots belong to a finite field, and, conversely, every nonzero element of a finite field is a root of unity. Any algebraically closed field contains exactly Template:Mvar Template:Mvarth roots of unity, except when Template:Mvar is a multiple of the (positive) characteristic of the field.

General definitionEdit

File:Visualisation complex number roots.svg

Geometric representation of the 2nd to 6th root of a general complex number in polar form. For the nth root of unity, set Template:Mvar = 1 and Template:Mvar = 0. The principal root is in black.

An Template:Mvarth root of unity, where Template:Mvar is a positive integer, is a number Template:Mvar satisfying the equation<ref>Template:Cite book</ref><ref>Template:Cite book</ref> <math display="block">z^n = 1. </math> Unless otherwise specified, the roots of unity may be taken to be complex numbers (including the number 1, and the number −1 if Template:Mvar is even, which are complex with a zero imaginary part), and in this case, the Template:Mvarth roots of unity are<ref name="meserve">Template:Cite book</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 (and even over any ring) Template:Math, and this allows considering roots of unity in Template:Math. Whichever is the field Template:Math, the roots of unity in Template:Math are either complex numbers, if the characteristic of Template:Math is 0, or, otherwise, belong to a finite field. Conversely, every nonzero element in a finite field is a root of unity in that field. See Root of unity modulo n and Finite field for further details.

An Template:Mvarth root of unity is said to be Template:Visible anchor if it is not an Template:Mvarth root of unity for some smaller Template:Mvar, that is if<ref name="moskowitz">Template:Cite book</ref><ref name="lidl">Template:Cite book</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 Template:Mathth roots of unity, except 1, are primitive.<ref name="morandi">Template:Cite book</ref>

In the above formula in terms of exponential and trigonometric functions, the primitive Template:Mvarth roots of unity are those for which Template:Mvar and Template:Mvar 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 Template:Slink. For the case of roots of unity in rings of modular integers, see Root of unity modulo n.

Elementary propertiesEdit

Every Template:Mathth root of unity Template:Math is a primitive Template:Mathth root of unity for some Template:Math, which is the smallest positive integer such that Template:Math.

Any integer power of an Template:Mathth root of unity is also an Template:Mathth root of unity,<ref name="integer-power">Template:Cite book</ref> as

This is also true for negative exponents. In particular, the reciprocal of an Template:Mathth root of unity is its complex conjugate, and is also an Template:Mathth root of unity:<ref name="conjugate">Template:Cite book</ref>

If Template:Math is an Template:Mathth root of unity and Template:Math then Template:Math. Indeed, by the definition of congruence modulo n, Template:Math for some integer Template:Math, and hence

Therefore, given a power Template:Math of Template:Math, one has Template:Math, where Template:Math is the remainder of the Euclidean division of Template:Mvar by Template:Mvar.

Let Template:Math be a primitive Template:Mathth root of unity. Then the powers Template:Math, Template:Math, ..., Template:Math, Template:Math are Template:Mathth roots of unity and are all distinct. (If Template:Math where Template:Math, then Template:Math, which would imply that Template:Math would not be primitive.) This implies that Template:Math, Template:Math, ..., Template:Math, Template:Math are all of the Template:Mathth roots of unity, since an Template:Mathth-degree polynomial equation over a field (in this case the field of complex numbers) has at most Template:Math solutions.

From the preceding, it follows that, if Template:Math is a primitive Template:Mathth root of unity, then <math>z^a = z^b</math> if and only if <math>a\equiv b \pmod{ n}.</math> If Template:Math 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 Template:Math, a non-primitive Template:Mathth root of unity is Template:Math, and one has <math>z^2 = z^4 = 1</math>, although <math> 2 \not\equiv 4 \pmod{4}.</math>

Let Template:Math be a primitive Template:Mathth root of unity. A power Template:Math of Template:Mvar is a primitive Template:Mathth root of unity for

where <math>\gcd(k,n)</math> is the greatest common divisor of Template:Mvar and Template:Mvar. This results from the fact that Template:Math is the smallest multiple of Template:Mvar that is also a multiple of Template:Mvar. In other words, Template:Math is the least common multiple of Template:Mvar and Template:Mvar. Thus

<math>a =\frac{\operatorname{lcm}(k,n)}{k}=\frac{kn}{k\gcd(k,n)}=\frac{n}{\gcd(k,n)}.</math>

Thus, if Template:Math and Template:Math are coprime, Template:Math is also a primitive Template:Mathth root of unity, and therefore there are Template:Math distinct primitive Template:Mathth roots of unity (where Template:Math is Euler's totient function). This implies that if Template:Math is a prime number, all the roots except Template:Math are primitive.

In other words, if Template:Math is the set of all Template:Mathth roots of unity and Template:Math is the set of primitive ones, Template:Math is a disjoint union of the Template:Math:

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

where the notation means that Template:Math goes through all the positive divisors of Template:Math, including Template:Math and Template:Math.

Since the cardinality of Template:Math is Template:Math, and that of Template:Math is Template:Math, this demonstrates the classical formula

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

Group propertiesEdit

Group of all roots of unityEdit

The product and the multiplicative inverse of two roots of unity are also roots of unity. In fact, if Template:Math and Template:Math, then Template:Math, and Template:Math, where Template:Math is the least common multiple of Template:Math and Template:Math.

Therefore, the roots of unity form an abelian group under multiplication. This group is the torsion subgroup of the circle group.

Group of Template:Mathth roots of unityEdit

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

Given a primitive Template:Mathth root of unity Template:Math, the other Template:Mathth roots are powers of Template:Math. This means that the group of the Template:Mathth 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 Template:Mathth roots of unityEdit

Let <math>\Q(\omega)</math> be the field extension of the rational numbers generated over <math>\Q</math> by a primitive Template:Mathth root of unity Template:Math. As every Template:Mathth root of unity is a power of Template:Math, the field <math>\Q(\omega)</math> contains all Template:Mathth roots of unity, and <math>\Q(\omega)</math> is a Galois extension of <math>\Q.</math>

If Template:Math is an integer, Template:Math is a primitive Template:Mathth root of unity if and only if Template:Math and Template:Math are coprime. In this case, the map

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

induces an automorphism of <math>\Q(\omega)</math>, which maps every Template:Mathth root of unity to its Template:Mathth 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 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 units of the ring of [[integers modulo n|integers modulo Template:Math]] and the Galois group of <math>\Q(\omega).</math>

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

Galois group of the real part of the primitive roots of unityEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The real part of the primitive roots of unity are related to one another as roots of the 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 expressionEdit

File:3rd roots of unity correction.svg

The cube roots of unity

De Moivre's formula, which is valid for all real Template:Mvar and integers Template:Mvar, is

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

Setting Template:Math gives a primitive Template:Mvarth 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 Template:Math. In other words,

is a primitive Template:Mvarth root of unity.

This formula shows that in the complex plane the Template:Mvarth roots of unity are at the vertices of a [[regular polygon|regular Template:Mvar-sided polygon]] inscribed in the unit circle, with one vertex at 1 (see the plot for Template:Math 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

which is valid for all real Template:Mvar, can be used to put the formula for the Template:Mvarth roots of unity into the form

It follows from the discussion in the previous section that this is a primitive Template:Mvarth-root if and only if the fraction Template:Math is in lowest terms; that is, that Template:Mvar and Template:Mvar are coprime. An irrational number that can be expressed as the real part of the root of unity; that is, as <math>\cos(2\pi k/n)</math>, is called a trigonometric number.

Algebraic expressionEdit

The Template:Mathth roots of unity are, by definition, the roots of the polynomial Template:Math, and are thus algebraic numbers. As this polynomial is not irreducible (except for Template:Math), the primitive Template:Mathth roots of unity are roots of an irreducible polynomial (over the integers) of lower degree, called the Template:Mathth cyclotomic polynomial, and often denoted Template:Math. The degree of Template:Math is given by Euler's totient function, which counts (among other things) the number of primitive Template:Mathth roots of unity.<ref name="riesel">Template:Cite book</ref> The roots of Template:Math are exactly the primitive Template:Mathth 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 Template:Mvar, there exists an expression built from integers by root extractions, additions, subtractions, multiplications, and divisions (and nothing else), such that the primitive Template:Mvarth roots of unity are exactly the set of values that can be obtained by choosing values for the root extractions (Template:Mvar possible values for a Template:Mvarth root). (For more details see Template:Slink, below.)

Gauss proved that a primitive Template:Mvarth root of unity can be expressed using only square roots, addition, subtraction, multiplication and division if and only if it is possible to construct with compass and straightedge the [[regular polygon|regular Template:Mvar-gon]]. This is the case if and only if Template:Math is either a power of two or the product of a power of two and Fermat primes that are all different.

If Template:Mvar is a primitive Template:Mvarth root of unity, the same is true for Template:Math, and <math>r=z+\frac 1z</math> is twice the real part of Template:Mvar. In other words, Template:Math is a reciprocal polynomial, the polynomial <math>R_n</math> that has Template:Mvar as a root may be deduced from Template:Math by the standard manipulation on reciprocal polynomials, and the primitive Template:Mvarth 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 Template:Mvar is a product of a power of two by a product (possibly empty) of distinct Fermat primes, and the regular Template:Mvar-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 degreesEdit

For Template:Math, the cyclotomic polynomial is Template:Math Therefore, the only primitive first root of unity is 1, which is a non-primitive Template:Mathth root of unity for every n > 1.
As Template:Math, the only primitive second (square) root of unity is −1, which is also a non-primitive Template:Mathth root of unity for every even Template:Math. With the preceding case, this completes the list of real roots of unity.
As Template:Math, the primitive third (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 Template:Math, the two primitive fourth roots of unity are Template:Math and Template:Math.
As Template:Math, 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 Template:Math, 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 roots. 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 Template:Mvar 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 Template:Math, the four primitive eighth roots of unity are the square roots of the primitive fourth roots, Template:Math. 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.

PeriodicityEdit

If Template:Mvar is a primitive Template:Mvarth root of unity, then the sequence of powers

Template:Math

is Template:Mvar-periodic (because Template:Math for all values of Template:Mvar), and the Template:Mvar sequences of powers

Template:Math

for Template:Math are all Template:Mvar-periodic (because Template:Math). Furthermore, the set Template:Math} of these sequences is a basis of the linear space of all Template:Mvar-periodic sequences. This means that any Template:Mvar-periodic sequence of complex numbers

Template:Math

can be expressed as a linear combination of powers of a primitive Template:Mvarth 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 Template:Math and every integer Template:Mvar.

This is a form of Fourier analysis. If Template:Mvar is a (discrete) time variable, then Template:Mvar is a frequency and Template:Math is a complex amplitude.

Choosing for the primitive Template:Mvarth root of unity

allows Template:Math to be expressed as a linear combination of Template:Math and Template:Math:

This is a discrete Fourier transform.

SummationEdit

Let Template:Math be the sum of all the Template:Mvarth 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 Template:Mvarth roots of unity being the roots of the polynomial Template:Math, their sum is the coefficient of degree Template:Math, which is either 1 or 0 according whether Template:Math or Template:Math.

Alternatively, for Template:Math there is nothing to prove, and for Template:Math there exists a root Template:Math – since the set Template:Math of all the Template:Mvarth roots of unity is a group, Template:Math, so the sum satisfies Template:Math, whence Template:Math.

Let Template:Math be the sum of all the primitive Template:Mvarth roots of unity. Then

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

where Template:Math is the Möbius function.

In the section Elementary properties, it was shown that if Template:Math is the set of all Template:Mvarth roots of unity and Template:Math is the set of primitive ones, Template:Math is a disjoint union of the Template:Math:

<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 Template:Math, then Template:Math, and for Template:Math: Template:Math. Therefore, Template:Math.

This is the special case Template:Math of Ramanujan's sum Template:Math,<ref name="apostol">Template:Cite book</ref> defined as the sum of the Template:Mvarth powers of the primitive Template:Mvarth roots of unity:

OrthogonalityEdit

From the summation formula follows an orthogonality relationship: for Template:Math and Template:Math

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

where Template:Mvar is the Kronecker delta and Template:Mvar is any primitive Template:Mvarth root of unity.

The Template:Math matrix Template:Mvar whose Template:Mathth entry is

defines a discrete Fourier transform. Computing the inverse transformation using Gaussian elimination requires Template:Math operations. However, it follows from the orthogonality that Template:Mvar is 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 Template:Mvar is simply the complex conjugate. (This fact was first noted by Gauss when solving the problem of trigonometric interpolation.) The straightforward application of Template:Mvar or its inverse to a given vector requires Template:Math operations. The fast Fourier transform algorithms reduces the number of operations further to Template:Math.

Cyclotomic polynomialsEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

The zeros of the polynomial

are precisely the Template:Mvarth roots of unity, each with multiplicity 1. The Template:Mvarth cyclotomic polynomial is defined by the fact that its zeros are precisely the primitive Template:Mvarth roots of unity, each with multiplicity 1.

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

where Template:Math are the primitive Template:Mvarth roots of unity, and Template:Math is Euler's totient function. The polynomial Template:Math 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 Template:Mvar, which is easier than the general assertion, follows by applying Eisenstein's criterion to the polynomial

and expanding via the binomial theorem.

Every Template:Mvarth root of unity is a primitive Template:Mvarth root of unity for exactly one positive divisor Template:Mvar of Template:Mvar. This implies that<ref name="riesel" />

This formula represents the factorization of the polynomial Template:Math 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 Template:Math is the Möbius function. So the first few cyclotomic polynomials are

Template:Math

If Template:Mvar is a prime number, then all the Template:Mvarth roots of unity except 1 are primitive Template:Mvarth 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 ≥ 2 for Template:Mvar, this sum becomes a [[radix|base Template:Mvar]] 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 Template:Math. It is not a surprise it takes this long to get an example, because the behavior of the coefficients depends not so much on Template:Mvar as on how many odd prime factors appear in Template:Mvar. More precisely, it can be shown that if Template:Mvar has 1 or 2 odd prime factors (for example, Template:Math) then the Template:Mvarth cyclotomic polynomial only has coefficients 0, 1 or −1. Thus the first conceivable Template:Mvar for which there could be a coefficient besides 0, 1, or −1 is a product of the three smallest odd primes, and that is Template:Math. 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 Template:Math occurs as a coefficient in the Template:Mvarth cyclotomic polynomial.<ref name="lehmer">Template:Cite journal</ref>

Many restrictions are known about the values that cyclotomic polynomials can assume at integer values. For example, if Template:Mvar is prime, then Template:Math if and only if Template:Math.

Cyclotomic polynomials are solvable in radicals, as roots of unity are themselves radicals. Moreover, there exist more informative radical expressions for Template:Mvarth roots of unity with the additional property<ref name="landau">Template:Cite journal</ref> that every value of the expression obtained by choosing values of the radicals (for example, signs of square roots) is a primitive Template:Mvarth root of unity. This was already shown by Gauss in 1797.<ref>Template:Cite book</ref> Efficient algorithms exist for calculating such expressions.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Cyclic groupsEdit

The Template:Mvarth roots of unity form under multiplication a cyclic group of order Template:Mvar, 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 Template:Mvarth root of unity.

The Template:Mvarth roots of unity form an irreducible representation of any cyclic group of order Template:Mvar. 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; 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">Template:Cite book</ref>Template:Page needed In particular, if a circulant Hermitian matrix is considered (for example, a discretized one-dimensional Laplacian with periodic boundaries<ref name="siam">Template:Cite journal</ref>), the orthogonality property immediately follows from the usual orthogonality of eigenvectors of Hermitian matrices.

Cyclotomic fieldsEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} By adjoining a primitive Template:Mvarth root of unity to <math>\Q,</math> one obtains the Template:Mvarth cyclotomic field <math>\Q(\exp(2\pi i/n)).</math>This field contains all Template:Mvarth roots of unity and is the splitting field of the Template:Mvarth 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 naturally 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 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.<ref>The Disquisitiones was published in 1801, 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 Kronecker, usually called the Kronecker–Weber theorem on the grounds that Weber completed the proof.

Relation to quadratic integersEdit

File:Roots of unity, golden ratio.svg

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.

File:Star polygon 8-2.svg

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

For Template:Math, both roots of unity Template:Num and Template:Num are integers.

For three values of Template:Mvar, the roots of unity are quadratic integers:

For Template:Math they are Eisenstein integers (Template:Math).
For Template:Math they are Gaussian integers (Template:Math): see Imaginary unit.

For four other values of Template:Mvar, the primitive roots of unity are not quadratic integers, but the sum of any root of unity with its complex conjugate (also an Template:Mvarth root of unity) is a quadratic integer.

For Template:Math, none of the non-real roots of unity (which satisfy a quartic equation) is a quadratic integer, but the sum Template:Math of each root with its complex conjugate (also a 5th root of unity) is an element of the ring [[quadratic integer|Template:Math]] (Template:Math). For two pairs of non-real 5th roots of unity these sums are inverse golden ratio and minus golden ratio.

For Template:Math, for any root of unity Template:Math equals to either 0, ±2, or ±[[square root of 2|Template:Sqrt]] (Template:Math).

For Template:Math, for any root of unity, Template:Math equals to either 0, ±1, ±2 or ±[[square root of 3|Template:Sqrt]] (Template:Math).

NotesEdit

Template:Reflist

ReferencesEdit

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Root of unity

Contents

General definitionEdit

Elementary propertiesEdit

Group propertiesEdit

Group of all roots of unityEdit

Group of Template:Mathth roots of unityEdit

Galois group of the primitive Template:Mathth roots of unityEdit

Galois group of the real part of the primitive roots of unityEdit

Trigonometric expressionEdit

Algebraic expressionEdit

Explicit expressions in low degreesEdit

PeriodicityEdit

SummationEdit

OrthogonalityEdit

Cyclotomic polynomialsEdit

Cyclic groupsEdit

Cyclotomic fieldsEdit

Relation to quadratic integersEdit

See alsoEdit

NotesEdit

ReferencesEdit