idele group of a number field

数域的伊代尔群

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

Other names / variants: idèle group

Translation evidence: Literature usage

Meaning in context

For a number field $K$, the restricted product of $K_v^\times$ relative to $\mathcal O_v^\times$ at finite places; algebraically it is $\mathbb A_K^\times$.

数域 $K$ 的各 $K_v^\times$ 关于有限素位处 $\mathcal O_v^\times$ 的限制直积；在代数上等于阿代尔环的可逆元群。

Example

The diagonal tuple $(2,2,2,\ldots)$ is an idele of $\mathbb Q$, with inverse $(1/2,1/2,1/2,\ldots)$.

Usage and distinctions

Nonzero components alone are insufficient: the tuple with component p at every finite prime p is an adele but not an idele.

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 3.48, 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3.48 伊代尔群，printed page 89 · Checked 2026-10-14

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