False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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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.

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Zp is the full profinite completion of the integers

Statement

$Z_{p}$ is the full profinite completion of $Z$ .

Facts & Assumptions

Given: A prime $p$ .

[L1]

$Z_{p}$ is the pro- $p$ completion of $Z$ (Zp is the pro-p completion of the integers).

[L2]

The full profinite completion of $Z$ is $\prod_{q} Z_{q}$ over all primes $q$ (The profinite completion of the integers is the direct product of the p-adic integer groups over all primes).

Refutation

technique · direct

1.1L1given

By [L1], $Z_{p}$ remembers only the finite quotients of $Z$ whose order is a power of $p$ .

2.1L2step 1.1

By [L2], the full profinite completion also has the $q$ -primary factor $Z_{q}$ for every prime $q \neq p$ . Therefore $Z_{p}$ omits the prime-to- $p$ information and cannot be the whole profinite completion.

3.1step 1.1step 2.1∎

So the stated claim is false.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

Jordan Bell, The profinite completion of the integers, the p-adic integers, and Prufer p-groups (standard reference, not scraped)