ExampleConstruction: Literature-sourcedVerification: 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.

A square root of minus one in Q_5

Example

The element $- 1$ is a square in $Q_{5}$ .

Facts & Assumptions

Given: The polynomial $f (X) = X^{2} + 1$ over $Z_{5}$ .

[L1]

For odd $p$ , a unit of $Z_{p}$ is a square in $Q_{p}$ exactly when its residue class is a square in $F_{p}^{\times}$ (Square criterion in Q_p for odd p).

[L2]

Simple roots lift uniquely, and Newton iteration gives the same root (Simple roots lift uniquely in Z_p, Newton's criterion in Q_p).

Verification

technique · direct

1.1

Modulo $5$ one has $2^{2} \equiv - 1$ . Since $- 1$ is a unit, [L1] shows that $- 1$ is a square in $Q_{5}$ .

L1givenalgebra

2.1

Concretely, $f (2) = 5 \equiv 0 (m o d 5)$ and $f^{'} (2) = 4 \equiv̸ 0 (m o d 5)$ , so [L2] produces a unique $5$ -adic root congruent to $2$ modulo $5$ .

L2step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

Keith Conrad, Hensel's Lemma, Examples 4.3 and 4.4 (standard reference, not scraped)