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.
Content and primitive integer polynomials
Definition
Let . Through the trimmed-list correspondence (Finitely supported coefficient sequences and trimmed finite coefficient lists define the same formal polynomials), write with . Its content is the iterated nonnegative integer gcd
using the integer gcd of Common divisor, and the greatest common divisor , with the convention . The result is positive because not every coefficient is zero. The polynomial is primitive when . Set , but do not call the zero polynomial primitive.
The value does not depend on appending trailing zero coefficients, and Content is the positive common divisor of the coefficients divisible by every common divisor ↗ proves its universal common-divisor characterization.
Depends on
- The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution
- Finitely supported coefficient sequences and trimmed finite coefficient lists define the same formal polynomials
- The integers form a commutative ring
- Common divisor, and the greatest common divisor $\gcd(a,b)$, with the convention $\gcd(0,0) := 0$
Used by
- Content is the positive common divisor of the coefficients divisible by every common divisor Lemma
- If p is a prime not dividing n, a rational minimal polynomial of a primitive n-th root of unity also kills its p-th power Lemma
- The product of primitive integer polynomials is primitive, and contents multiply Lemma
- Φ_pʳ(t)=∑_k<pt^kpʳ⁻¹, and Φ_pʳ(t+1) is Eisenstein at p Proposition
Dependency tree · two levels
24 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
- Keith Conrad, Irreducibility Tests in Q[T], Appendix A.1 (standard reference, not scraped)