# Dirichlet’s unit theorem

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Let $K$ be a number field^{}, and let ${\mathcal{O}}_{K}$ be its ring of integers^{}.
Then

$${\mathcal{O}}_{K}^{*}\cong \mu (K)\times {\mathbb{Z}}^{r+s-1}.$$ |

Here ${\mathcal{O}}_{K}^{*}$ is the group of units of ${\mathcal{O}}_{K}$,
$\mu (K)$ is the finite cyclic group of the roots of unity^{} in ${\mathcal{O}}_{K}^{*}$,
$r$ is the number of real embeddings $K\to \mathbb{R}$,
and $2s$ is the number of non-real complex embeddings $K\to \u2102$ (which occur in complex conjugate^{} pairs, so $s$ is an integer).

Title | Dirichlet’s unit theorem |
---|---|

Canonical name | DirichletsUnitTheorem |

Date of creation | 2013-03-22 13:22:42 |

Last modified on | 2013-03-22 13:22:42 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 10 |

Author | yark (2760) |

Entry type | Theorem |

Classification | msc 11R04 |

Classification | msc 11R27 |

Related topic | Regulator^{} |