# adèle

Let $K$ be a number field^{}. For each finite prime $v$ of $K$, let
${\U0001d52c}_{v}$ denote the valuation ring^{} of the completion ${K}_{v}$ of $K$ at
$v$. The adèle group ${\mathbb{A}}_{K}$ of $K$ is defined to be the
restricted direct product^{} of the collection of locally compact
additive groups^{} $\{{K}_{v}\}$ over all primes $v$ of $K$ (both finite
primes and infinite primes), with respect to the collection of compact^{}
open subgroups $\{{\U0001d52c}_{v}\}$ defined for all finite primes $v$.

The set ${\mathbb{A}}_{K}$ inherits addition and multiplication operations (defined pointwise) which make it into a topological ring. The original field $K$ embeds as a ring into ${\mathbb{A}}_{K}$ via the map

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

defined for $x\in K$, where ${x}_{v}$ denotes the image of $x$ in ${K}_{v}$ under the embedding $K\hookrightarrow {K}_{v}$. Note that ${x}_{v}\in {\U0001d52c}_{v}$ for all but finitely many $v$, so that the element $x$ is sent under the above definition into the restricted direct product as claimed.

It turns out that the image of $K$ in ${\mathbb{A}}_{K}$ is a discrete set and the
quotient group^{} ${\mathbb{A}}_{K}/K$ is a compact space in the quotient topology.

Title | adèle |
---|---|

Canonical name | Adele |

Date of creation | 2013-03-22 12:39:31 |

Last modified on | 2013-03-22 12:39:31 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 5 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 11R56 |

Related topic | Idele |

Defines | adèle group |

Defines | group of adèles |