p-adic number field

p 进数域

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

Other names / variants: field of p-adic numbers

Translation evidence: Literature usage

Meaning in context

For prime $p$, $\mathbb Q_p$ is the completion of $\mathbb Q$ for the absolute value $|x|_p=p^{-v_p(x)}$.

对素数 $p$，$\mathbb Q_p$ 是有理数域关于绝对值 $|x|_p=p^{-v_p(x)}$ 的完备化。

Example

In $\mathbb Q_3$, $1+3+3^2+\cdots=-1/2$.

Usage and distinctions

Despite its name, $\mathbb Q_p$ is not a number field in the finite-extension-of-$\mathbb Q$ sense.

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, Proposition prop:p-adic and following paragraph defining Q_p as completion of Q · Checked 2026-10-15

Translation sources

1. 李文威 — 代数学方法 卷一，第五章，作者源文件命题 prop:p-adic 后的 p-进数域 定义段落 · Checked 2026-10-15

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