polynomial-proof

What problem does it solve?

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.

Core Features & Use Cases

• Eisenstein-based irreducibility checks: apply Eisenstein's criterion and its substitutions to establish irreducibility over common rings.
• Reduction mod p techniques: use modulo-p reductions to infer irreducibility over fields of characteristic zero from finite fields.
• Factor theorem and rational root tests: identify linear factors and possible decompositions, with guidance on higher-degree cases.
• Minimal polynomials: compute or identify minimal polynomials for algebraic numbers and verify irreducibility.

Quick Start

Provide a polynomial and the base ring, and the skill will walk you through irreducibility checks, factorization, and minimal polynomials.