Alphabeta MathDefinitionDefinition: 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.
Irreducible and prime elements of an integral domain
Definition
Let be an integral domain and let be nonzero and not a unit. The element is irreducible if every factorisation has or a unit. It is prime if
for all .
Depends on
Used by
- An irreducible polynomial over a field is separable exactly when its derivative is nonzero Corollary
- Unique factorisation domain Definition
- Every nonzero nonunit polynomial over a field factors into irreducible polynomials Lemma
- Every prime element of an integral domain is irreducible Lemma
- A polynomial of degree two or three over a field is irreducible exactly when it has no root in the field Theorem
- Every irreducible polynomial over a field is prime Theorem
- For a nonconstant p in F[x], the ideal (p) is maximal and F[x]/(p) is a field exactly when p is irreducible Theorem
- Gauss lemma: primitive factorisations over ℚ can be cleared to primitive factorisations over ℤ Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 12 results over 11 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)