DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-03

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.

Unique factorisation domain

Definition

An integral domain $R$ is a unique factorisation domain (UFD) when every nonzero nonunit is a finite product of irreducible elements, and whenever

$a=p_1\cdots p_m=q_1\cdots q_n$

are products of irreducibles, then $m=n$ and, after a permutation, each $p_i$ is associate to the corresponding $q_i$ . Thus uniqueness is only up to order and multiplication of factors by units.

Depends on

Used by

For every field F, F[x] is a unique factorisation domain Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 10 results over 9 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Sharifi, Abstract Algebra, Advanced Ring Theory (standard reference, not scraped)