Math 2.0

Number theory Theorem

Number of nth roots of unity

Builds on The nth roots of unity

Statement

Let nn be a positive integer. Then there are exactly nn complex nnth roots of unity.

Remarks

The list of the nth roots of unity has nn numbers e2πik/ne^{2 \pi i k / n} with 0≤k<n0 \le k < n. They are pairwise different, because their arguments are different numbers in [0,2π)[0, 2\pi). A polynomial of degree nn has at most nn roots in a field, so Xn−1X^n - 1 has no other roots. In other fields there can be fewer: in ℝ\mathbb{R}, the only 44th roots of unity are 11 and −1-1.

Reading path

To understand this page from the ground up, read these 18 pages in order. The foundations come first.

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Complex numbers, object, no evidence levelComplex numbersobjectReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionEuler's formula, statement, no evidence levelEuler's formulastatementProduct of exponentials, statement, no evidence levelProduct of exponentia…statementReal and imaginary parts, definition, no evidence levelReal and imaginary pa…definitionRoot of unity, definition, no evidence levelRoot of unitydefinitionSupremum, definition, no evidence levelSupremumdefinitionde Moivre's formula, statement, no evidence levelde Moivre's formulastatementCompleteness axiom, statement, no evidence levelCompleteness axiomstatementExistence of square roots of nonnegative reals, statement, no evidence levelExistence of square r…statementSquare root, definition, no evidence levelSquare rootdefinitionModulus of a complex number, definition, no evidence levelModulus of a complex …definitionArgument of a complex number, definition, no evidence levelArgument of a complex…definitionPrincipal value of the argument, statement, no evidence levelPrincipal value of th…statementPolar form, statement, no evidence levelPolar formstatementThe nth roots of unity, statement, no evidence levelThe nth roots of unitystatementNumber of nth roots of unity, statement, no evidence levelNumber of nth roots o…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Complex numbers object
  2. Real numbers object
  3. Set bounded above definition
  4. Euler's formula statement
  5. Product of exponentials statement
  6. Real and imaginary parts definition
  7. Root of unity definition
  8. Supremum definition
  9. de Moivre's formula statement
  10. Completeness axiom statement
  11. Existence of square roots of nonnegative reals statement
  12. Square root definition
  13. Modulus of a complex number definition
  14. Argument of a complex number definition
  15. Principal value of the argument statement
  16. Polar form statement
  17. The nth roots of unity statement
  18. Number of nth roots of unity statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Root_of_unity, Elementary properties
About this page
Tag0E021GF2B8
URIm2://0E021GF2B8
Kindstatement, theorem
WritingW0 stub
MSC11R18
StatusThis copy of the site has no build data. Run m2 status 0E021GF2B8.
UsesThe nth roots of unity

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