In a lot of places, I can read that the roots of unity form a cyclic group, however I can find no proofs. Is the reasoning as follows:(adsbygoogle = window.adsbygoogle || []).push({});

Let's work in a field of characteristic zero (I think that's necessary). Let's look at the nth roots of unity, i.e. the solutions of [itex]x^n - 1[/itex]. There are n different roots, since the derivative is [itex]nx^{n-1}[/itex], which is not zero since the characteristic is zero. Now suppose the group of roots isnotcyclic, then the exponent of that group is [itex]m < n [/itex]. In that case the group is also the set of solutions of [itex]x^m-1[/itex], however this can only have m solutions. Contradiction.

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Roots of unity form a cyclic group

Loading...

Similar Threads - Roots unity form | Date |
---|---|

Roots of unity, Roots of complex equations of the form z^n = 1 | Nov 18, 2014 |

Galois Groups of Extensions by Roots of Unity | Apr 24, 2012 |

Roots of unity of matrices | Dec 5, 2009 |

Is e^ix multivalued, roots of unity, etc | Jan 9, 2009 |

C10 Group, 10th roots unity with complex number multiplication | Sep 23, 2008 |

**Physics Forums - The Fusion of Science and Community**