adele ring of a number field

数域的阿代尔环

Number Theory · Graduate · Mathematics Resources editorial compilation (AI-assisted)

Other names / variants: adèle ring

Translation evidence: Literature usage

Meaning in context

For a number field $K$, the ring of tuples $(x_v)\in\prod_v K_v$ such that $x_v\in\mathcal O_v$ at all but finitely many finite places, with the restricted product topology.

数域 $K$ 的各完备化 $K_v$ 的限制直积：元组 $(x_v)$ 在除有限多个有限素位外均属于 $\mathcal O_v$，并赋予限制直积拓扑。

Example

For $K=\mathbb Q$, $(0,0,0,\ldots)$ and the diagonal tuple $(1,1,1,\ldots)$ are adeles.

Usage and distinctions

The full product allows nonintegral components at infinitely many finite places; the adele ring imposes the almost-everywhere integrality condition.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. 张神星 — 代数数论讲义 v2.5 (2024-01-09)，合肥工业大学作者讲义Section 3.3.1, Definition 3.47, printed page 89 · Checked 2026-10-14

Translation sources

1. 张神星 — 代数数论讲义 v2.5 (2024-01-09)，合肥工业大学作者讲义定义3.47 阿代尔环，printed page 89 · Checked 2026-10-14

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