LemmaStatement: Literature-sourcedProof: 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.

Monicity and degree stay fixed during Hensel factor lifting

Statement

Let $A$ be a commutative ring, let $g, h \in A [T]$ be monic of degrees $m, n$ , and let $u, v \in A [T]$ satisfy $\deg u < m$ and $\deg v < n$ . Then $g + u$ and $h + v$ are still monic of degrees $m$ and $n$ .

Facts & Assumptions

Given: Monic polynomials $g, h$ of degrees $m, n$ and correction terms $u, v$ with $\deg u < m$ and $\deg v < n$ .

[L1]

A polynomial of degree less than $m$ has zero $T^{m}$ -coefficient, and similarly a polynomial of degree less than $n$ has zero $T^{n}$ -coefficient.

Proof

technique · compare leading terms

1.1

Since $\deg u < m$ , [L1] shows that the coefficient of $T^{m}$ in $u$ is $0$ . Hence the coefficient of $T^{m}$ in $g + u$ is the coefficient of $T^{m}$ in $g$ , namely $1$ . Therefore $g + u$ is monic of degree $m$ .

L1given

1.2

Since $\deg v < n$ , [L1] shows that the coefficient of $T^{n}$ in $v$ is $0$ . Hence the coefficient of $T^{n}$ in $h + v$ is the coefficient of $T^{n}$ in $h$ , namely $1$ . Therefore $h + v$ is monic of degree $n$ .

L1given

2.1

Hence degree-bounded corrections preserve the prescribed monicity and degrees throughout Hensel lifting.

step 1.1step 1.2∎

Used by

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Allen B. Altman and Steven L. Kleiman, A Term of Commutative Algebra, 13th ed., Chapter 22 (standard reference, not scraped)