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In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 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.

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 n nth roots of unity, except when n is a multiple of the (positive) characteristic of the field.

General definition

An nth root of unity, where n is a positive integer, is a number z satisfying the equation^[1][2] $${\displaystyle z^{n}=1.}$$ Unless otherwise specified, the roots of unity may be taken to be complex numbers (including the number 1, and the number −1 if n is even, which are complex with a zero imaginary part), and in this case, the nth roots of unity are^[3] $${\displaystyle \exp \left({\frac {2k\pi i}{n}}\right)=\cos {\frac {2k\pi }{n}}+i\sin {\frac {2k\pi }{n}},\qquad k=0,1,\dots ,n-1.}$$

However, the defining equation of roots of unity is meaningful over any field (and even over any ring) F, and this allows considering roots of unity in F. Whichever is the field F, the roots of unity in F are either complex numbers, if the characteristic of F 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 nth root of unity is said to be primitive if it is not an mth root of unity for some smaller m, that is if^[4][5]

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

If n is a prime number, then all nth roots of unity, except 1, are primitive.^[6]

In the above formula in terms of exponential and trigonometric functions, the primitive nth roots of unity are those for which k and n are coprime integers.

Subsequent sections of this article will comply with complex roots of unity. For the case of roots of unity in fields of nonzero characteristic, see Finite field § Roots of unity. For the case of roots of unity in rings of modular integers, see Root of unity modulo n.

Elementary properties

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

Any integer power of an nth root of unity is also an nth root of unity,^[7] as

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

This is also true for negative exponents. In particular, the reciprocal of an nth root of unity is its complex conjugate, and is also an nth root of unity:^[8]

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

If z is an nth root of unity and a ≡ b (mod n) then z^a = z^b. Indeed, by the definition of congruence modulo n, a = b + kn for some integer k, and hence

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

Therefore, given a power z^a of z, one has z^a = z^r, where 0 ≤ r < n is the remainder of the Euclidean division of a by n.

Let z be a primitive nth root of unity. Then the powers z, z², ..., zⁿ⁻¹, zⁿ = z⁰ = 1 are nth roots of unity and are all distinct. (If z^a = z^b where 1 ≤ a < b ≤ n, then z^b−a = 1, which would imply that z would not be primitive.) This implies that z, z², ..., zⁿ⁻¹, zⁿ = z⁰ = 1 are all of the nth roots of unity, since an nth-degree polynomial equation over a field (in this case the field of complex numbers) has at most n solutions.

From the preceding, it follows that, if z is a primitive nth root of unity, then ${\displaystyle z^{a}=z^{b}}$ if and only if ${\displaystyle a\equiv b{\pmod {n}}.}$ If z is not primitive then ${\displaystyle a\equiv b{\pmod {n}}}$ implies ${\displaystyle z^{a}=z^{b},}$ but the converse may be false, as shown by the following example. If n = 4, a non-primitive nth root of unity is z = –1, and one has ${\displaystyle z^{2}=z^{4}=1}$, although ${\displaystyle 2\not \equiv 4{\pmod {4}}.}$

Let z be a primitive nth root of unity. A power w = z^k of z is a primitive ath root of unity for

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

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

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

Thus, if k and n are coprime, z^k is also a primitive nth root of unity, and therefore there are φ(n) distinct primitive nth roots of unity (where φ is Euler's totient function). This implies that if n is a prime number, all the roots except +1 are primitive.

In other words, if R(n) is the set of all nth roots of unity and P(n) is the set of primitive ones, R(n) is a disjoint union of the P(n):

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

where the notation means that d goes through all the positive 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

Zdroj:https://en.wikipedia.org?pojem=Primitive_nth_root_of_unity
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