Proving Primitive Root of Unity: z = cos(2pi/n) + isin(2pi/n)

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SUMMARY

The discussion centers on proving that the expression z = cos(2π/n) + isin(2π/n) represents a primitive nth root of unity. The user seeks clarification on whether assuming z^k is not primitive implies that z must be an nth root of unity. The conversation emphasizes the importance of definitions for nth roots and primitive nth roots of unity, suggesting that employing a proof by contradiction may simplify the argument. The reference to De Moivre's Theorem indicates its relevance in the proof process.

PREREQUISITES
  • Understanding of complex numbers and their representation in polar form.
  • Familiarity with the definitions of nth roots and primitive nth roots of unity.
  • Knowledge of De Moivre's Theorem and its applications.
  • Basic proof techniques, particularly proof by contradiction.
NEXT STEPS
  • Study the definitions and properties of primitive roots of unity.
  • Learn about De Moivre's Theorem and its implications in complex analysis.
  • Explore proof techniques, especially proof by contradiction, in mathematical arguments.
  • Investigate the relationship between roots of unity and their geometric interpretations on the complex plane.
USEFUL FOR

Students of mathematics, particularly those studying complex analysis, algebra, or number theory, will benefit from this discussion. It is especially relevant for anyone looking to understand roots of unity and their properties in mathematical proofs.

Driessen12
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Homework Statement



show that cos(2pi/n) + isin(2pi/n) is a primitive root of unity

Homework Equations





The Attempt at a Solution


if i know z = cos(2pi/n) + isin(2pi/n) is an nth root and I'm trying to prove that z is a primitive nth root. is it correct to assume that z^k is not primitive and is therefore an nth root of unity. I just need to know if z is an nth root and i know z^k is not primitive, does it have to be an nth root? just want to make sure there are no subtleties. I have the rest of the proof. I think this is easier if I use contradiction and that would require me to assume that if z^k is not primitive that it is an nth root.
 
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I think it will be quite helpful (to you) to state the precise definitions of (n-th) root and primitive (n-th) root of unity. Also Moivre[/url] can be useful here.
 
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