Determine polynomial irreducibility and factorizations over specified rings.
npx skills add https://github.com/0bserver07/bourbaki --skill polynomial-proof
Please help me install this Agent Skill. Skill: polynomial-proof Source: https://github.com/0bserver07/bourbaki/tree/main/src/skills/polynomial-proof Command: npx skills add https://github.com/0bserver07/bourbaki --skill polynomial-proof
This Skill helps mathematicians and students reason about when polynomials are irreducible, how to factor them, and how to determine minimal polynomials across different coefficient rings, with structured criteria and examples.
Provide a polynomial and the base ring, and the skill will walk you through irreducibility checks, factorization, and minimal polynomials.
High-intent search queries and answers about installing and using this skill.
To prove polynomial irreducibility, apply Eisenstein's criterion, reduction mod p, or the Factor Theorem. Provide the polynomial and base ring to receive a structured verdict, potential factors, and references to the specific methods used in the proof.
Reduction mod p tests polynomial irreducibility by reducing coefficients modulo a prime and checking the polynomial over a finite field. If the reduced polynomial is irreducible, the original is irreducible over fields of characteristic zero.
Yes, Eisenstein's criterion and its substitutions establish irreducibility over common rings. The skill applies these checks to input polynomials with integer or polynomial coefficients, outputting a structured verdict on irreducibility.
To find the minimal polynomial for an algebraic number, provide the number and context to compute or identify the polynomial and verify its irreducibility. The skill outputs the minimal polynomial result suitable for symbolic computation or Lean verification.
The best way to factor polynomials over integer coefficients is using the Factor Theorem and rational root tests to identify linear factors and possible decompositions. The skill provides potential factors with guidance on higher-degree cases.
Eisenstein's criterion should not be used when no prime satisfies the required divisibility conditions on the polynomial's coefficients. In such cases, alternative methods like reduction mod p or the Factor Theorem are needed to determine irreducibility or factorization.
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