CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck 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.

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

A local ring is Henselian exactly when simple residue roots lift uniquely

Statement

Let $(A, m)$ be a local ring. Then $A$ is Henselian if and only if every simple root of every monic polynomial over the residue field $A / m$ lifts uniquely to a root over $A$ .

Facts & Assumptions

Given: A local ring $(A, m)$ .

[L1]

Henselian factor lifting implies unique simple-root lifting (Factor lifting implies simple-root lifting).

[L2]

Unique simple-root lifting recovers the coprime factor-lifting property (The simple-root form recovers factor lifting).

[L3]

A Henselian local ring is precisely a local ring whose maximal-ideal pair is Henselian (Henselian pairs and Henselian local rings).

Proof

technique · prove the two implications separately

1.1

If $A$ is Henselian, then [L3] identifies $(A, m)$ as a Henselian pair, and [L1] gives unique lifting of every simple residue root.

L1L3given

1.2

Conversely, assume every simple residue root lifts uniquely. Then [L2] gives the coprime monic factor-lifting property for $(A, m)$ . Since $m$ is the unique maximal ideal of the local ring, one has $m \subseteq J (A)$ , so [L3] shows that $A$ is Henselian.

L2L3given

2.1

Therefore a local ring is Henselian exactly when every simple residue root lifts uniquely.

step 1.1step 1.2∎

Depends on

Used by

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

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