CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passjudge 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.

Z_p is the valuation ring of Q_p

Statement

Under the comparison of The p-adic completion agrees with the fraction field of Z_p, the $(p)$ -adic completion $Z_{p}$ of $Z$ identifies with

${x \in Q_{p} : ∣ x ∣_{p} \leq 1} .$

Facts & Assumptions

Given: The canonical embedding $Z_{p} ↪ Q_{p}$ .

[L1]

The comparison theorem identifies $Q_{p}$ with the fraction field of $Z_{p}$ and identifies the embedded copy of $Z_{p}$ with the closed unit ball (The p-adic completion agrees with the fraction field of Z_p).

Proof

technique · direct

1.1

By [L1], an element of $Q_{p}$ lies in the image of $Z_{p}$ exactly when its $p$ -adic absolute value is at most $1$ .

L1given

2.1

Thus the image is exactly ${x \in Q_{p} : ∣ x ∣_{p} \leq 1}$ , which is the valuation ring of the valued field $Q_{p}$ .

step 1.1∎

Depends on

The p-adic completion agrees with the fraction field of Z_p

Used by

Qₚ is locally compact and totally disconnected Corollary
Simple roots lift uniquely in Zₚ Corollary
The maximal ideal and residue field of Zₚ Corollary
Zₚ is not the integral closure of Z in Qₚ Counterexample
Every p-adic number has a unique digit expansion Theorem
Newton's criterion in Qₚ Theorem
Square criterion in Q₂ Theorem
Square criterion in Qₚ for odd p Theorem

Dependency tree · two levels

7 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 8, Remark 8.2 (standard reference, not scraped)
J. S. Milne, Algebraic Number Theory, Chapter 7 (standard reference, not scraped)