p-adic additive valuation

p 进加性赋值

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

Other names / variants: p-adic order

Translation evidence: Literature usage

Meaning in context

For prime $p$ and nonzero $x\in\mathbb Q$, write $x=p^e a/b$ with $p\nmid ab$ and define $v_p(x)=e$; set $v_p(0)=+\infty$.

对素数 $p$，若非零有理数 $x=p^e a/b$ 且 $p\nmid ab$，则 $v_p(x)=e$；约定 $v_p(0)=+\infty$。

Example

$v_3(18/5)=2$ and $v_3(1/9)=-2$.

Usage and distinctions

This additive exponent is different from the multiplicative absolute value $|x|_p=p^{-v_p(x)}$.

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Definition 2.3 and Exercise 2.1.4, printed page 32; rational specialization · 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2.3 加性赋值及习题2.1.4 p进赋值，printed page 32 · Checked 2026-10-14

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