ring of integers of a number field

数域的整数环

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

Other names / variants: number-field integer ring

Translation evidence: Literature usage

Meaning in context

For a number field $K$, $\mathcal O_K$ is the subring of elements integral over $\mathbb Z$.

对数域 $K$，其整数环 $\mathcal O_K$ 是 $K$ 中所有在 $\mathbb Z$ 上整的元素组成的子环。

Example

$\mathcal O_{\mathbb Q(\sqrt5)}=\mathbb Z[(1+\sqrt5)/2]$.

Usage and distinctions

It is the full integral closure, and may be larger than a ring generated by a convenient algebraic integer.

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Definitions 1.10, 1.14 and Example 1.2, printed pages 6–7 · 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§1.2.2 定义1.14 整数环，printed page 7 · Checked 2026-10-14

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