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 truncated polynomial local ring is Henselian

Example

Let $k$ be a field and let $A = k [x] / (x^{n}) (n \geq 1) .$ Then $A$ is a local Artinian ring, hence Henselian.

Facts & Assumptions

Given: A field $k$ , an integer $n \geq 1$ , and the quotient ring $A = k [x] / (x^{n})$ .

[L1]

Polynomial rings and quotient rings are the ambient objects in which this example lives (The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution, The quotient ring $R / I$ with $(r + I) (s + I) = r s + I$ ).

[L2]

Artinian local rings are Henselian (Artinian local rings are Henselian).

Verification

technique · identify the maximal ideal and its nilpotence

1.1

In $A$ , every class with nonzero constant term is a unit, so the nonunits are exactly the classes divisible by $x$ . Thus $A$ is local with maximal ideal $(x)$ , and $(x)^{n} = 0$ .

L1givenalgebra

2.1

The descending chain of ideals in $A$ is finite because every ideal is one of $(1), (x), \dots, (x^{n - 1}), (0)$ , so $A$ is Artinian. Therefore [L2] applies and shows that $A$ is Henselian.

L2step 1.1algebra

3.1

Hence the truncated polynomial local ring $k [x] / (x^{n})$ is a concrete Artinian Henselian ring.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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

The Stacks Project, Section 10.153: Henselian local rings (standard reference, not scraped)