# idèle

Let $K$ be a number field^{}. For each finite prime $v$ of $K$, let ${\U0001d52c}_{v}$ be the valuation ring^{} of the completion ${K}_{v}$ of $K$ at $v$, and let ${U}_{v}$ be the group of units in ${\U0001d52c}_{v}$. Then each group ${U}_{v}$ is a compact^{} open subgroup of the group of units ${K}_{v}^{*}$ of ${K}_{v}$. The idèle group ${\mathbb{I}}_{K}$ of $K$ is defined to be the restricted direct product^{} of the multiplicative groups^{} $\{{K}_{v}^{*}\}$ with respect to the compact open subgroups $\{{U}_{v}\}$, taken over all finite primes and infinite primes $v$ of $K$.

The units ${K}^{*}$ in $K$ embed into ${\mathbb{I}}_{K}$ via the diagonal embedding

$$x\mapsto \prod _{v}{x}_{v},$$ |

where ${x}_{v}$ is the image of $x$ under the embedding $K\hookrightarrow {K}_{v}$ of $K$ into its completion ${K}_{v}$. As in the case of adèles, the group ${K}^{*}$ is a discrete subgroup of the group of idèles ${\mathbb{I}}_{K}$, but unlike the case of adèles, the quotient group^{} ${\mathbb{I}}_{K}/{K}^{*}$ is not a compact group. It is, however, possible to define a certain subgroup^{} of the idèles (the subgroup of norm 1 elements) which does have compact quotient under ${K}^{*}$.

Warning: The group ${\mathbb{I}}_{K}$ is a multiplicative subgroup of the ring of adèles ${\mathbb{A}}_{K}$, but the topology^{} on ${\mathbb{I}}_{K}$ is different from the subspace topology that ${\mathbb{I}}_{K}$ would have as a subset of ${\mathbb{A}}_{K}$.

Title | idèle |
---|---|

Canonical name | Idele |

Date of creation | 2013-03-22 12:39:28 |

Last modified on | 2013-03-22 12:39:28 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 7 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 11R56 |

Related topic | Adele |

Defines | idèle group |

Defines | group of idèles |