HOME

TheInfoList



OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a root of unity, occasionally called a
de Moivre Abraham de Moivre FRS (; 26 May 166727 November 1754) was a French mathematician known for de Moivre's formula, a formula that links complex numbers and trigonometry, and for his work on the normal distribution and probability theory. He mov ...
number, is any
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
that yields 1 when raised to some positive integer power . 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 mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a comple ...
. 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 In logic and mathematics, the converse of a categorical or implicational statement is the result of reversing its two constituent statements. For the implication ''P'' → ''Q'', the converse is ''Q'' → ''P''. For the categorical proposit ...
, every nonzero element of a finite field is a root of unity. Any algebraically closed field contains exactly th roots of unity, except when is a multiple of the (positive) characteristic of the field.


General definition

An ''th root of unity'', where is a positive integer, is a number satisfying the equation :z^n = 1. Unless otherwise specified, the roots of unity may be taken to be
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s (including the number 1, and the number −1 if is
even Even may refer to: General * Even (given name), a Norwegian male personal name * Even (surname) * Even (people), an ethnic group from Siberia and Russian Far East **Even language, a language spoken by the Evens * Odd and Even, a solitaire game wh ...
, which are complex with a zero imaginary part), and in this case, the th roots of unity are :\exp\left(\frac\right)=\cos\frac+i\sin\frac,\qquad k=0,1,\dots, n-1. However, the defining equation of roots of unity is meaningful over any field (and even over any ring) , and this allows considering roots of unity in . Whichever is the field , the roots of unity in are either complex numbers, if the characteristic of 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 th root of unity is said to be if it is not an th root of unity for some smaller , that is if :z^n=1\quad \text \quad z^m \ne 1 \text m = 1, 2, 3, \ldots, n-1. If ''n'' is a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
, then all th roots of unity, except 1, are primitive. In the above formula in terms of exponential and trigonometric functions, the primitive th roots of unity are those for which and are coprime integers. Subsequent sections of this article will comply with complex roots of unity. For the case of roots of unity in fields of nonzero characteristic, see . For the case of roots of unity in rings of modular integers, see Root of unity modulo ''n''.


Elementary properties

Every th root of unity is a primitive th root of unity for some , which is the smallest positive integer such that . Any integer power of an th root of unity is also an th root of unity, as :(z^k)^n = z^ = (z^n)^k = 1^k = 1. This is also true for negative exponents. In particular, the
reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another pol ...
of an th root of unity is its complex conjugate, and is also an th root of unity: :\frac = z^ = z^ = \bar z. If is an th root of unity and then . Indeed, by the definition of congruence modulo ''n'', for some integer , and hence : z^a = z^ = z^b z^ = z^b (z^n)^k = z^b 1^k = z^b. Therefore, given a power of , one has , where is the remainder of the Euclidean division of by . Let be a primitive th root of unity. Then the powers , , ..., , are th roots of unity and are all distinct. (If where , then , which would imply that would not be primitive.) This implies that , , ..., , are all of the th roots of unity, since an th- degree polynomial equation over a field (in this case the field of complex numbers) has at most solutions. From the preceding, it follows that, if is a primitive th root of unity, then z^a = z^b
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
a\equiv b \pmod. If is not primitive then a\equiv b \pmod implies z^a = z^b, but the converse may be false, as shown by the following example. If , a non-primitive th root of unity is , and one has z^2 = z^4 = 1, although 2 \not\equiv 4 \pmod. Let be a primitive th root of unity. A power of is a primitive th root of unity for : a = \frac, where \gcd(k,n) is the greatest common divisor of and . This results from the fact that is the smallest multiple of that is also a multiple of . In other words, is the least common multiple of and . Thus :a =\frac=\frac=\frac. Thus, if and are coprime, is also a primitive th root of unity, and therefore there are distinct primitive th roots of unity (where is Euler's totient function). This implies that if is a prime number, all the roots except are primitive. In other words, if is the set of all th roots of unity and is the set of primitive ones, is a disjoint union of the : :\operatorname(n) = \bigcup_\operatorname(d), where the notation means that goes through all the positive
divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
s of , including and . Since the cardinality of is , and that of is , this demonstrates the classical formula :\sum_\varphi(d) = n.


Group properties


Group of all roots of unity

The product and the
multiplicative inverse In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/' ...
of two roots of unity are also roots of unity. In fact, if and , then , and , where is the least common multiple of and . Therefore, the roots of unity form an
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
under multiplication. This group is the
torsion subgroup In the theory of abelian groups, the torsion subgroup ''AT'' of an abelian group ''A'' is the subgroup of ''A'' consisting of all elements that have finite order (the torsion elements of ''A''). An abelian group ''A'' is called a torsion group (or ...
of the circle group.


Group of th roots of unity

For an integer ''n'', the product and the multiplicative inverse of two th roots of unity are also th roots of unity. Therefore, the th roots of unity form an abelian group under multiplication. Given a primitive th root of unity , the other th roots are powers of . This means that the group of the th roots of unity is a cyclic group. 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 th roots of unity

Let \Q(\omega) be the field extension of the rational numbers generated over \Q by a primitive th root of unity . As every th root of unity is a power of , the field \Q(\omega) contains all th roots of unity, and \Q(\omega) is a Galois extension of \Q. If is an integer, is a primitive th root of unity if and only if and are coprime. In this case, the map :\omega \mapsto \omega^k induces an automorphism of \Q(\omega), which maps every th root of unity to its th power. Every automorphism of \Q(\omega) is obtained in this way, and these automorphisms form the Galois group of \Q(\omega) 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 :k\mapsto \left(\omega \mapsto \omega^k\right) defines a group isomorphism between the units of the ring of integers modulo and the Galois group of \Q(\omega). This shows that this Galois group is
abelian Abelian may refer to: Mathematics Group theory * Abelian group, a group in which the binary operation is commutative ** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms * Metabelian group, a grou ...
, and implies thus that the primitive roots of unity may be expressed in terms of
radicals Radical may refer to: Politics and ideology Politics *Radical politics, the political intent of fundamental societal change *Radicalism (historical), the Radical Movement that began in late 18th century Britain and spread to continental Europe and ...
.


Trigonometric expression

De Moivre's formula, which is valid for all real and integers , is :\left(\cos x + i \sin x\right)^n = \cos nx + i \sin nx. Setting gives a primitive th root of unity – one gets :\left(\cos\frac + i \sin\frac\right)^ = \cos 2\pi + i \sin 2\pi = 1, but :\left(\cos\frac + i \sin\frac\right)^ = \cos\frac + i \sin\frac \neq 1 for . In other words, :\cos\frac + i \sin\frac is a primitive th root of unity. This formula shows that in the complex plane the th roots of unity are at the vertices of a regular -sided polygon inscribed in the unit circle, with one vertex at 1 (see the plots for and on the right). This geometric fact accounts for the term "cyclotomic" in such phrases as cyclotomic field and cyclotomic polynomial; it is from the Greek roots " cyclo" (circle) plus " tomos" (cut, divide).
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that ...
:e^ = \cos x + i \sin x, which is valid for all real , can be used to put the formula for the th roots of unity into the form :e^, \quad 0 \le k < n. It follows from the discussion in the previous section that this is a primitive th-root if and only if the fraction is in lowest terms; that is, that and are coprime. An irrational number that can be expressed as the real part of the root of unity; that is, as \cos(2\pi k/n), is called a trigonometric number.


Algebraic expression

The th roots of unity are, by definition, the
roots A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients. Root or roots may also refer to: Art, entertainment, and media * ''The Root'' (magazine), an online magazine focusing ...
of the polynomial , and are thus algebraic numbers. As this polynomial is not irreducible (except for ), the primitive th roots of unity are roots of an irreducible polynomial of lower degree, called the th cyclotomic polynomial, and often denoted . The degree of is given by Euler's totient function, which counts (among other things) the number of primitive th roots of unity. The roots of are exactly the primitive th roots of unity. Galois theory can be used to show that cyclotomic polynomials may be conveniently solved in terms of radicals. (The trivial form \sqrt /math> is not convenient, because it contains non-primitive roots, such as 1, which are not roots of the cyclotomic polynomial, and because it does not give the real and imaginary parts separately.) This means that, for each positive integer , there exists an expression built from integers by root extractions, additions, subtractions, multiplications, and divisions (and nothing else), such that the primitive th roots of unity are exactly the set of values that can be obtained by choosing values for the root extractions ( possible values for a th root). (For more details see , below.) Gauss proved that a primitive th root of unity can be expressed using only
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . ...
s, addition, subtraction, multiplication and division if and only if it is possible to construct with compass and straightedge the regular -gon. This is the case
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
is either a
power of two A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer  as the exponent. In a context where only integers are considered, is restricted to non-negat ...
or the product of a power of two and Fermat primes that are all different. If is a primitive th root of unity, the same is true for , and r=z+\frac 1z is twice the real part of . In other words, is a reciprocal polynomial, the polynomial R_n that has as a root may be deduced from by the standard manipulation on reciprocal polynomials, and the primitive th roots of unity may be deduced from the roots of R_n by solving the
quadratic equation In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not qu ...
z^2-rz+1=0. That is, the real part of the primitive root is \frac r2, and its imaginary part is \pm i\sqrt. The polynomial R_n is an irreducible polynomial whose roots are all real. Its degree is a power of two, if and only if is a product of a power of two by a product (possibly
empty Empty may refer to: ‍ Music Albums * ''Empty'' (God Lives Underwater album) or the title song, 1995 * ''Empty'' (Nils Frahm album), 2020 * ''Empty'' (Tait album) or the title song, 2001 Songs * "Empty" (The Click Five song), 2007 * ...
) of distinct Fermat primes, and the regular -gon is constructible with compass and straightedge. Otherwise, it is solvable in radicals, but one are in the casus irreducibilis, that is, every expression of the roots in terms of radicals involves ''nonreal radicals''.


Explicit expressions in low degrees

* For , the cyclotomic polynomial is Therefore, the only primitive first root of unity is 1, which is a non-primitive th root of unity for every ''n'' > 1. * As , the only primitive second (square) root of unity is −1, which is also a non-primitive th root of unity for every even . With the preceding case, this completes the list of real roots of unity. * As , the primitive third ( cube) roots of unity, which are the roots of this quadratic polynomial, are \frac,\ \frac . * As , the two primitive fourth roots of unity are and . * As , the four primitive fifth roots of unity are the roots of this
quartic polynomial In algebra, a quartic function is a function of the form :f(x)=ax^4+bx^3+cx^2+dx+e, where ''a'' is nonzero, which is defined by a polynomial of degree four, called a quartic polynomial. A ''quartic equation'', or equation of the fourth deg ...
, which may be explicitly solved in terms of radicals, giving the roots \frac4 \pm i \frac, where \varepsilon may take the two values 1 and −1 (the same value in the two occurrences). * As , there are two primitive sixth roots of unity, which are the negatives (and also the square roots) of the two primitive cube roots: \frac,\ \frac. * As 7 is not a Fermat prime, the seventh roots of unity are the first that require cube 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 r^3+r^2-2r-1, and the primitive seventh roots of unity are \frac\pm i\sqrt, where runs over the roots of the above polynomial. As for every cubic polynomial, these roots may be expressed in terms of square and cube roots. However, as these three roots are all real, this is casus irreducibilis, and any such expression involves non-real cube roots. * As , the four primitive eighth roots of unity are the square roots of the primitive fourth roots, . They are thus \pm\frac \pm i\frac. * See Heptadecagon for the real part of a 17th root of unity.


Periodicity

If is a primitive th root of unity, then the sequence of powers : is -periodic (because for all values of ), and the sequences of powers : for are all -periodic (because ). Furthermore, the set of these sequences is a basis of the linear space of all -periodic sequences. This means that ''any'' -periodic sequence of complex numbers : can be expressed as a linear combination of powers of a primitive th root of unity: : x_j = \sum_k X_k \cdot z^ = X_1 z^ + \cdots + X_n \cdot z^ for some complex numbers and every integer . This is a form of Fourier analysis. If is a (discrete) time variable, then is a
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
and is a complex
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
. Choosing for the primitive th root of unity :z = e^\frac = \cos\frac + i \sin\frac allows to be expressed as a linear combination of and : :x_j = \sum_k A_k \cos \frac + \sum_k B_k \sin \frac. This is a
discrete Fourier transform In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a comple ...
.


Summation

Let be the sum of all the th roots of unity, primitive or not. Then :\operatorname(n) = \begin 1, & n=1\\ 0, & n>1. \end This is an immediate consequence of
Vieta's formulas In mathematics, Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. They are named after François Viète (more commonly referred to by the Latinised form of his name, "Franciscus Vieta"). Basic formula ...
. In fact, the th roots of unity being the roots of the polynomial , their sum is the coefficient of degree , which is either 1 or 0 according whether or . Alternatively, for there is nothing to prove, and for there exists a root – since the set of all the th roots of unity is a group, , so the sum satisfies , whence . Let be the sum of all the primitive th roots of unity. Then :\operatorname(n) = \mu(n), where is the Möbius function. In the section Elementary properties, it was shown that if is the set of all th roots of unity and is the set of primitive ones, is a disjoint union of the : :\operatorname(n) = \bigcup_\operatorname(d), This implies :\operatorname(n) = \sum_\operatorname(d). Applying the Möbius inversion formula gives :\operatorname(n) = \sum_\mu(d)\operatorname\left(\frac\right). In this formula, if , then , and for : . Therefore, . This is the special case of Ramanujan's sum , defined as the sum of the th powers of the primitive th roots of unity: :c_n(s) = \sum_^n e^.


Orthogonality

From the summation formula follows an orthogonality relationship: for and :\sum_^ \overline \cdot z^ = n \cdot\delta_ where is the Kronecker delta and is any primitive th root of unity. The
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
whose th entry is :U_ = n^\cdot z^ defines a
discrete Fourier transform In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a comple ...
. Computing the inverse transformation using Gaussian elimination requires operations. However, it follows from the orthogonality that is
unitary Unitary may refer to: Mathematics * Unitary divisor * Unitary element * Unitary group * Unitary matrix * Unitary morphism * Unitary operator * Unitary transformation * Unitary representation In mathematics, a unitary representation of a grou ...
. That is, :\sum_^ \overline \cdot U_ = \delta_, and thus the inverse of is simply the complex conjugate. (This fact was first noted by Gauss when solving the problem of trigonometric interpolation). The straightforward application of or its inverse to a given vector requires operations. The fast Fourier transform algorithms reduces the number of operations further to .


Cyclotomic polynomials

The zeros of the polynomial :p(z) = z^n - 1 are precisely the th roots of unity, each with multiplicity 1. The th '' cyclotomic polynomial'' is defined by the fact that its zeros are precisely the ''primitive'' th roots of unity, each with multiplicity 1. : \Phi_n(z) = \prod_^(z-z_k) where are the primitive th roots of unity, and is Euler's totient function. The polynomial has integer coefficients and is an irreducible polynomial over the rational numbers (that is, it cannot be written as the product of two positive-degree polynomials with rational coefficients). The case of prime , which is easier than the general assertion, follows by applying Eisenstein's criterion to the polynomial :\frac, and expanding via the binomial theorem. Every th root of unity is a primitive th root of unity for exactly one positive
divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
of . This implies that :z^n - 1 = \prod_ \Phi_d(z). This formula represents the factorization of the polynomial into irreducible factors: :\begin z^1 -1 &= z-1 \\ z^2 -1 &= (z-1)(z+1) \\ z^3 -1 &= (z-1) (z^2 + z + 1) \\ z^4 -1 &= (z-1)(z+1) (z^2+1) \\ z^5 -1 &= (z-1) (z^4 + z^3 +z^2 + z + 1) \\ z^6 -1 &= (z-1)(z+1) (z^2 + z + 1) (z^2 - z + 1)\\ z^7 -1 &= (z-1) (z^6+ z^5 + z^4 + z^3 + z^2 + z + 1) \\ z^8 -1 &= (z-1)(z+1) (z^2+1) (z^4+1) \\ \end Applying Möbius inversion to the formula gives :\Phi_n(z) = \prod_\left(z^\frac - 1\right)^ = \prod_\left(z^d - 1\right)^, where is the Möbius function. So the first few cyclotomic polynomials are : : : : : : : : If is a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
, then all the th roots of unity except 1 are primitive th roots, and we have : \Phi_p(z) = \frac = \sum_^ z^k. Substituting any positive integer ≥ 2 for , this sum becomes a base repunit. Thus a necessary (but not sufficient) condition for a repunit to be prime is that its length be prime. Note that, contrary to first appearances, ''not'' all coefficients of all cyclotomic polynomials are 0, 1, or −1. The first exception is . It is not a surprise it takes this long to get an example, because the behavior of the coefficients depends not so much on as on how many
odd Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric. Odd may also refer to: Acronym * ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...
prime factors appear in . More precisely, it can be shown that if has 1 or 2 odd prime factors (for example, ) then the th cyclotomic polynomial only has coefficients 0, 1 or −1. Thus the first conceivable for which there could be a coefficient besides 0, 1, or −1 is a product of the three smallest odd primes, and that is . This by itself doesn't prove the 105th polynomial has another coefficient, but does show it is the first one which even has a chance of working (and then a computation of the coefficients shows it does). A theorem of Schur says that there are cyclotomic polynomials with coefficients arbitrarily large in absolute value. In particular, if n = p_1 p_2 \cdots p_t, where p_1 < p_2 < \cdots < p_t are odd primes, p_1 +p_2>p_t, and ''t'' is odd, then occurs as a coefficient in the th cyclotomic polynomial. Many restrictions are known about the values that cyclotomic polynomials can assume at integer values. For example, if is prime, then if and only . Cyclotomic polynomials are solvable in
radicals Radical may refer to: Politics and ideology Politics *Radical politics, the political intent of fundamental societal change *Radicalism (historical), the Radical Movement that began in late 18th century Britain and spread to continental Europe and ...
, as roots of unity are themselves radicals. Moreover, there exist more informative radical expressions for th roots of unity with the additional property that every value of the expression obtained by choosing values of the radicals (for example, signs of square roots) is a primitive th root of unity. This was already shown by Gauss in 1797. Efficient
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
s exist for calculating such expressions.


Cyclic groups

The th roots of unity form under multiplication a cyclic group of
order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
, 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 th root of unity. The th roots of unity form an irreducible representation of any cyclic group of order . 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. In particular, if a circulant Hermitian matrix is considered (for example, a discretized one-dimensional Laplacian with periodic boundaries), the orthogonality property immediately follows from the usual orthogonality of eigenvectors of Hermitian matrices.


Cyclotomic fields

By adjoining a primitive th root of unity to \Q, one obtains the th cyclotomic field \Q(\exp(2\pi i/n)).This field contains all th roots of unity and is the splitting field of the th cyclotomic polynomial over \Q. The field extension \Q(\exp(2\pi 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(2\pi 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 ''n''th root of unity may be expressed in term of ''k''-roots, with various ''k'' not exceeding φ(''n''). In these cases Galois theory can be written out explicitly in terms of Gaussian periods: this theory from the '' Disquisitiones Arithmeticae'' of Gauss was published many years before Galois.The ''Disquisitiones'' was published in 1801, Galois was born in 1811, died in 1832, but wasn't published until 1846. Conversely, ''every'' abelian extension of the rationals is such a subfield of a cyclotomic field – this is the content of a theorem of Kronecker, usually called the '' Kronecker–Weber theorem'' on the grounds that Weber completed the proof.


Relation to quadratic integers

For , both roots of unity and are integers. For three values of , the roots of unity are quadratic integers: * For they are Eisenstein integers (). * For they are Gaussian integers (): see Imaginary unit. For four other values of , the primitive roots of unity are not quadratic integers, but the sum of any root of unity with its complex conjugate (also an th root of unity) is a quadratic integer. For , none of the non-real roots of unity (which satisfy a quartic equation) is a quadratic integer, but the sum of each root with its complex conjugate (also a 5th root of unity) is an element of the ring Z /a>(). For two pairs of non-real 5th roots of unity these sums are
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when a ...
golden ratio and
minus The plus and minus signs, and , are mathematical symbols used to represent the notions of positive and negative, respectively. In addition, represents the operation of addition, which results in a sum, while represents subtraction, resul ...
golden ratio. For , for any root of unity equals to either 0, ±2, or ± (). For , for any root of unity, equals to either 0, ±1, ±2 or ± ().


See also

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


Notes


References

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