Alphabeta MathHow 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
- The reduction of Φₙ is irreducible over F_q exactly when [q] generates (ℤ/n)^× Corollary
- The p-primary component of a module over a domain Definition
- Unique factorisation domain Definition
- Every irreducible element of a principal ideal domain is prime Lemma
- Every nonzero nonunit polynomial over a field factors into irreducible polynomials Lemma
- Every prime element of an integral domain is irreducible Lemma
- Gauss lemma over a UFD Lemma
- Noetherian domains are atomic Lemma
- Φ_pʳ(t)=∑_k<pt^kpʳ⁻¹, and Φ_pʳ(t+1) is Eisenstein at p Proposition
- Φₙ is irreducible over K exactly when [K(ζₙ):K]=φ(n), exactly when the embedding into (ℤ/n)^× is onto Proposition
- 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
- For gcd(n,q)=1 the reduction of Φₙ in F_q[t] is a product of distinct monic irreducibles, each of degree the order of [q] modulo n Theorem
- Gauss lemma: primitive factorisations over ℚ can be cleared to primitive factorisations over ℤ Theorem
- Φₙ is irreducible in ℚ[t] for every n≥1 Theorem
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
- Sharifi, Abstract Algebra, Advanced Ring Theory (standard reference, not scraped)