# cyclotomic units

Let $L=\mathbb{Q}({\zeta}_{m})$ be a cyclotomic extension of $\mathbb{Q}$ with $m$ chosen to be minimal. Then the ring of integers^{} is given by $\mathbb{Z}({\zeta}_{m})$, and we denote the group of units by ${\mathcal{O}}_{L}^{\times}$. *The cyclotomic units* are the elements of a subgroup $C$ of ${\mathcal{O}}_{L}^{\times}$ given by

$\eta ={\displaystyle \frac{{\zeta}_{m}^{r}-1}{{\zeta}_{m}^{s}-1}}$ |

with $r$ and $s$ relatively prime to $m$.

Title | cyclotomic units |
---|---|

Canonical name | CyclotomicUnits |

Date of creation | 2013-03-22 14:12:36 |

Last modified on | 2013-03-22 14:12:36 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 5 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 11R18 |