Cyclotomy (surgery)

In mathematics, a root of unity is any complex number 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. 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 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 numbers (including the number 1, and the number −1 if is even, 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, 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 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 $a\equiv b \pmod.$
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 divisors 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 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 under multiplication. This group is the torsion subgroup 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, 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 unity

The real part of the primitive roots of unity are related to one another as roots of the minimal polynomial of $2\cos(2\pi/n).$ The roots of the minimal polynomial are just twice the real part; these roots form a cyclic Galois group.

Trigonometric expression

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 plot for 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

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