Math 2.0

Field theory and polynomials Theorem

Existence of nth roots of complex numbers

Builds on Polar formNumber of nth roots of unity

Statement

Let zz be a complex number and let nn be a positive integer. Then there is a complex number ww with wn=zw^n = z.

Remarks

For z=reiθz = r e^{i \theta} with r>0r > 0, one solution is w=r1/neiθ/nw = r^{1/n} e^{i \theta / n}, by the polar form. For z≠0z \neq 0, there are exactly nn solutions. They are ww times the nnth roots of unity. This fact is a special case of the fundamental theorem of algebra.

Reading path

To understand this page from the ground up, read these 19 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…statementExistence of nth roots of complex numbers, statement, no evidence levelExistence of nth root…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
  19. Existence of nth roots of complex numbers statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Nth_root, Complex roots
  2. Web https://en.wikipedia.org/wiki/Complex_number, Powers and roots
About this page
Tag0YM6P6C206
URIm2://0YM6P6C206
Kindstatement, theorem
WritingW0 stub
MSC12D99
StatusThis copy of the site has no build data. Run m2 status 0YM6P6C206.
UsesPolar form, Number of nth roots of unity

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