PropositionStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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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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Lifted coprime factorisations are unique

Statement

Let $(A, I)$ be a Henselian pair. Let $f \in A [T]$ be monic, and let $\overline{f} = g_{0} h_{0}$ in $(A / I) [T]$ with $g_{0}, h_{0}$ monic and coprime. Then there is at most one factorization $f = g h$ with $g, h$ monic and $\overline{g} = g_{0}$ , $\overline{h} = h_{0}$ .

Facts & Assumptions

Given: A Henselian pair $(A, I)$ , a monic polynomial $f \in A [T]$ , and a coprime monic residue factorization $\overline{f} = g_{0} h_{0}$ .

[L1]

In a Henselian pair, such lifted factorisations are part of the defining lifting property (Henselian pairs and Henselian local rings).

[L2]

Any two lifts of the same coprime residue factorization agree modulo every power of the ideal (Two lifted factorisations agree modulo every ideal power).

Proof

technique · combine the definition with the congruence induction

1.1

Suppose $f = g h = g^{'} h^{'}$ are two monic lifts of the given residue factorization. By [L2], they agree modulo $I^{r} [T]$ for every $r \geq 1$ .

L2given

2.1

In the present page's convention, [L1] already includes uniqueness of the lifted factorization. Therefore the two lifts must coincide. Step 1.1 records the explicit congruence mechanism that later examples use.

L1step 1.1

3.1

Hence a coprime monic residue factorization has at most one Hensel lift.

step 2.1∎

Depends on

Used by

Factor lifting implies simple-root lifting Corollary

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

Allen B. Altman and Steven L. Kleiman, A Term of Commutative Algebra, 13th ed., Chapter 22 (standard reference, not scraped)
The Stacks Project, Section 15.11: Henselian pairs (standard reference, not scraped)