ring of p-adic integers

p 进整数环

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

Other names / variants: p-adic integer ring

Translation evidence: Literature usage

Meaning in context

For prime $p$, $\mathbb Z_p=\{x\in\mathbb Q_p:v_p(x)\ge0\}$, equivalently the inverse limit of $\mathbb Z/p^n\mathbb Z$ under reduction.

对素数 $p$，$\mathbb Z_p=\{x\in\mathbb Q_p:v_p(x)\ge0\}$；等价地，它是 $\mathbb Z/p^n\mathbb Z$ 关于约化映射的逆极限。

Example

$1/2\in\mathbb Z_3$ but $1/3\notin\mathbb Z_3$.

Usage and distinctions

$\mathbb Z_p$ is a complete ring with infinitely many digits, not the finite quotient $\mathbb Z/p\mathbb Z$.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. 李文威 — 代数学方法 卷一，第五章，作者源文件Example eg:Z_p, discussion defining v_p and Proposition prop:p-adic · Checked 2026-10-15

Translation sources

1. 李文威 — 代数学方法 卷一，第五章，作者源文件例 eg:Z_p 中 p-进整数环；命题 prop:p-adic · Checked 2026-10-15

Original encyclopedia and terminology content: CC BY-SA 4.0. Contribute or report a correction.