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ID: INPUT{id.yaml}

Title: >
  Roots of unity

Definition: >
  The list contains roots of unity, that is, 
  complex solutions of equations of the form $x^n = 1$.
  The $n$'th roots of unity are given as $\exp(2\pi i k/n)$.

Parameters:
  n:
    type: Z
    constraints: $n \geq 1$
  k:
    type: Z
    constraints: $0\leq k < n$
   
Comments:
  comment-primitive-root-of-unity: >
    If $\gcd(k,n) = 1$, $\exp(2\pi i k/n)$ is a called primitive $n$'th root of unity.
    These are the roots of the $n$'th cyclotomic polynomial CITE{WikiCycPoly}.
      
Formulas:

Programs:

References:

Links:
  WikiCycPoly:
    title: "Wikipedia: Cyclotomic polynomial"
    url: https://en.wikipedia.org/wiki/Cyclotomic_polynomial
  Wiki:
    title: "Wikipedia: Root of unity"
    url: https://en.wikipedia.org/wiki/Root_of_unity
    
Similar tables:
  
Keywords:
  
Tags:
- zero
- algebraic

Data properties:
  type: C
  complete: no

Display properties:
  number-header: $\exp(2\pi i k/n)$

Numbers: INPUT{numbers.yaml}

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Roots of unity

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Numbers

$n$

$k$

$\exp(2\pi i k/n)$

INPUT{numbers.yaml} (not shown in preview)

Definition

The list contains roots of unity, that is, complex solutions of equations of the form $x^n = 1$. The $n$'th roots of unity are given as $\exp(2\pi i k/n)$.

Parameters

$n$

— integer ($n \geq 1$)

$k$

— integer ($0\leq k < n$)

Comments

(1)

If $\gcd(k,n) = 1$, $\exp(2\pi i k/n)$ is a called primitive $n$'th root of unity. These are the roots of the $n$'th cyclotomic polynomial [1].

Links

[1]

Wikipedia: Cyclotomic polynomial

[2]

Wikipedia: Root of unity

Data properties

Entries are of type: complex number

Table is complete: no