DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-04

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The p-adic absolute value on the rationals

Definition

Let $p$ be a prime. For $x \in Q$ define

$∣ 0 ∣_{p} : = 0, ∣ x ∣_{p} : = p^{- v_{p} (x)} (x \neq 0),$

where $v_{p}$ is the rational $p$ -adic valuation of The $p$ -adic valuation extends to the nonzero rationals by $v_{p} (a / b) : = v_{p} (a) - v_{p} (b) \in Z$ , independently of the representation; it satisfies $v_{p} (x y) = v_{p} (x) + v_{p} (y)$ , and $v_{p} (x + y) \geq \min {v_{p} (x), v_{p} (y)}$ whenever $x$ , $y$ and $x + y$ are nonzero. In particular,

$∣ p ∣_{p} = p^{- 1} .$

Depends on

The $p$-adic valuation extends to the nonzero rationals by $v_p(a/b) := v_p(a) - v_p(b) \in \mathbb{Z}$, independently of the representation; it satisfies $v_p(xy) = v_p(x) + v_p(y)$, and $v_p(x+y) \ge \min\{v_p(x), v_p(y)\}$ whenever $x$, $y$ and $x+y$ are nonzero

Used by

The p-adic numbers as a metric completion Definition
A geometric series that is p-adically convergent and really divergent Example
The same sequence behaves oppositely in the real and p-adic metrics Example
Ostrowski's theorem for the rationals Theorem
The p-adic absolute value is nonarchimedean Theorem
The product formula for the rationals Theorem

Dependency tree · two levels

22 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Andrew V. Sutherland, 18.782 Lecture 5 (standard reference, not scraped)
J. S. Milne, Algebraic Number Theory, Chapter 7 (standard reference, not scraped)